# Metamorphosis of Planar Parametric Curves via Curvature Interpolation

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Metamorphosis of Planar Parametric Curves via Curvature Interpolation Tatiana Surazhsky and Gershon Elber Technion, Israel Institute of Technology, Haifa 32000, Israel. Abstract This work considers the problem of metamorphosis interpolation between two freeform planar curves. Given two planar parametric curves, the curvature signature of the two curves is linearly blended, yielding a gradual change that is not only smooth but also employs intrinsic curvature shape properties, and hence is highly appealing. In order to be able to employ this curvature blending, we present a constructive scheme to derive curvature signatures of parameter curves. Additionally, we propose a scheme to reconstruct a curve from its curvature signature. Keywords: freeform curves, metamorphosis, animation, curvature analysis. 1 Introduction Morphing/metamorphing or the gradual and continuous transformation of one shape into another is a topic of great importance in computer graphics. Solutions to the metamorphosis problem have been presented under many domains, including two-dimensional images [2, 9, 20], planar polygons and polylines (i.e. piecewise linear curves) [10,14,15,17,18], polyhedra [1], free form curves [5,13,16], and even volumetric representations [6, 11]. The solution to the metamorphosis problem necessitates the resolution of two independent tasks. The first requires the derivation of a correspondence between the beginning and end objects. The second must deal with a proper continuous and appealing interpolation between the two shapes. In the context of piecewise linear planar curves (polylines or polygons), the correspondence problem is typically defined as a vertex correspondence problem. In other words, each vertex in the first key polygon is matched against one vertex (or more) in the second key polygon and vice versa (see [15] for an example). For higher order polynomial and rational planar curves, matching can be established as an optimization problem. For example, in [5] one of the two key curves is reparameterized so that the tangent fields of the two curves are made as parallel as possible. The second problem, as yet lacking any better solution, is solved in many cases via a linear interpolation approach. This approach has a fundamental flaw that may result in intermediate results that are counterintuitive, distorted, shrunken or even self-intersecting. In [14, 15], the bending and stretching energy is Center for Graphics and Geometric Computing, Applied Mathematics Department, Center for Graphics and Geometric Computing, Faculty of Computer Science, 1

2 evaluated and employed to optimize the interpolation between piecewise linear curves. Unfortunately, and while typically achieving a more appealing metamorphosis sequence, the work of [14, 15] cannot guarantee self-intersecting free intermediate results. The same approach is used in [16] for piecewise polynomial curves. The algorithm pesented in [16] linearly interpolates the B-spline curves that are matched by minimizing bending, stretching and kinking work functions in corresponding knot values of the key curves. The authors point to other interpolation options for the matched B-spline curves, such as [14]. As stated, linear interpolation of the control polygons and control meshes of freeform curves and surfaces is a common practice. Interestingly, linear interpolation of the derivatives of the two beginning and end freeforms yields no better results. Let C 1 (r) and C 2 (r) be two sufficiently differentiable parametric curves that share the same domain, and denote by C (d) and (d) C the d th derivative and integral of C, respectively. Then, the intermediate curve C(r) at time t that resulted from interpolating the d th derivatives of C 1 (r) and C 2 (r) equals C(r) = (d) C (d) 1 (r)(1 t) + C(d) 2 (r)tdt (d) (d) (d) = (1 t) C 1 (r)dt + t (d) C 2 (r)dt = (1 t)c 1 (r) + tc 2 (r), (1.1) up to the integrations constants. Specifically, interpolating the second order derivatives as an approximation to curvature interpolation yields little advantage and is identical to the interpolation of the curves themselves. In this work, we seek a smooth and appealing interpolation scheme between two freeform parametric curves. By linearly interpolating intrinsic curvature properties, the result is far more appealing and intuitive than the regular linear interpolation scheme of the curve or its derivatives. The fundamental theorem of differential geometry of curves shows us that the curvature field, κ(s), of a planar curve parameterized by arc-length C(s), fully prescribes it [3] up to a rigid motion transformation. In Section 2, we show how can one uniquely derive the curvature signature of a planar curve and more importantly, how can one reconstruct the curve C(s) from its curvature signature κ(s). The rest of the paper is organized as follows. Following the curvature reconstruction scheme that is discussed in Section 2, we present in Section 3 a metamorphosis scheme that interpolates two curvature signatures. In Section 4 we present examples of metamorphosis sequences between freeform curves employing the proposed scheme, and finally, we conclude in Section 5. 2

3 2 Curve Reconstruction Using Curvature Signatures Let C(s) be an arc-length planar parametric curve. Then C (s) = κ(s)n(s), N(s) being the unit normal field of C(s). The fundamental theorem of differential geometry of curves [3] states that κ(s) fully prescribes planar curve C(s) up to a rigid motion transformation. That is, two planar curves share the same curvature signature κ(s) if and only if one is a rigid motion transformation of the other. In Section 2.1, we consider the derivation of κ(s) from C(s) while in Section 2.2, the reconstruction of C(s) from κ(s) is investigated. 2.1 Computation of the Curvature Signature Given a non arc-length planar curve, C(t) = (x(t), y(t)), the derivation of the curvature field of κ(t) is fairly simple [3]: assuming C(t) is regular or C (t) 0. κ(t) = x (t)y (t) x (t)y (t) (x 2 (t) + y 2 (t)) 3/2, (2.1) Polynomial and rational forms cannot, in general, be reparameterized as arc-length parametric forms. Yet, one can arbitrary closely approximate t(s), a reparameterization function from t to the arc-length parameter s. In [8], an algorithm is presented that constructs an arc-length polynomial approximation to a given polynomial curve. This algorithm can be made arbitrarily precise, using refinement [4]. Hence, methods to approximate t(s) are both known and tractable. Once t(s) is approximated to within a certain tolerance, κ(s) = κ(t(s)) can be evaluated as a composition [8] between piecewise polynomial functions, to yield κ(s) = x (s)y (s) x (s)y (s), (2.2) because ( x 2 (s) + y 2 (s) ) = 1. The composition of polynomials [8] can be computed analytically and has a fixed complexity that depends on the orders of the two curves and lengths of their control polygons. 2.2 Reconstruction of a Curve from its Curvature Signature The more difficult question we are about to examine is how can one reconstruct C(s), given κ(s). From differential geometry, we know that C (s) = T (s), C (s) = T (s) = κ(s)n(s), 3

4 where T (s) and N(s) are the unit tangent and normal fields of C(s). T (s) = (x (s), y (s)) is a unit size vector and hence is always on the unit circle. Let θ be the angle between T (s) and the x-axis, ( y θ(s) = tan 1 ) (s) x (s) and consider θ (s), θ (s) = = ( ( y tan 1 )) (s) x (s) ( y (s) x (s) ) 2 ( y (s) x (s) x 2 (s) x (s)y (s) x (s)y (s) = x 2 (s) + y 2 (s) x 2 (s) = x (s)y (s) x (s)y (s) x 2 (s) + y 2 (s) ) = x (s)y (s) x (s)y (s), again since ( x 2 (s) + y 2 (s) ) = 1. But then (compare with Equation (2.2)), θ (s) = κ(s). Hence, assuming θ(0) = ϕ 0, θ(s) = s 0 κ( s)d s + ϕ 0. The assumption of θ(0) = ϕ 0 simply states that the reconstructed curve will be oriented so that its tangent s angle is ϕ 0 at s = 0, pinning down the degree of freedom of the orientation of the rigid motion. κ(s) is a scalar field. In order to derive C(s), which is a vector-valued function, we need to introduce an arc-length unit circle with which κ(s) will be composed. Let Circ(s), s [0, 2π] be an arc-length parameterized unit circle, Circ(s) = (cos(s), sin(s)). Then, T (s) = Circ(θ(s)) = Circ ( s 0 κ( s)d s + ϕ 0) prescribes the unit tangent field of C(s). Hence, we can finally write C(s) = = s 0 s 0 T ( s)d s Circ ( s 0 ) ( ) κ( s)d s + ϕ 0 d s + rx, 0 ry 0. (2.3) This result is known (see, for example, page 24, question 9 of [3]). Yet, we now examine the computational aspects of Equation (2.3) starting with the two of the simplest possible cases of reconstructing the curve from its curvature signature, thus getting a grasp of the expected complexity. In Section 2.3, we consider the case of κ(s) = 1 and in Section 2.4 we look at κ(s) = s. Then, in Section 2.5, we will consider the computational aspects of evaluating Equation (2.3) for a general polynomial curvature signature. 2.3 Reconstructing the Curve of κ(s) = 1 Substituting 1 for κ( s) in Equation (2.3) one gets s ( s ) C 1 (s) = Circ 1 d s d s 4

5 (0,1) Figure 1: The curve of κ(s) = 1, ϕ 0 = 0 and (r 0 x, r 0 y) = (0, 0). Figure 2: The curve of κ(s) = s, ϕ 0 = 0 and (r 0 x, r 0 y) = (0, 0). = = s Circ ( s + ϕ0 ) d s s(cos( s + ϕ0 ), sin( s + ϕ 0 ))d s = (sin(s + ϕ 0 ) + r 0 x, cos(s + ϕ 0 ) + r 0 y). (2.4) It is not surprising that the curve with a curvature of constant one turned out to be the unit circle itself (see Figure 1). ϕ 0 is the gained rotation degree of freedom. Moreover, any translation (r x, r y ) of C 1 (r) in the plane is also a valid solution for κ(s) = Reconstructing the Curve of κ(s) = s Substituting s for κ( s) in Equation (2.3) one gets C s (s) = s Circ ( s s d s ) d s 5

6 = = ( ) s s 2 Circ 2 + ϕ 0 d s ( ( ) ( )) s s 2 s 2 cos 2 + ϕ 0, sin 2 + ϕ 0 d s. (2.5) It is quite unfortunate but even for this simple case of κ(s) = s, the resulting curve (See Figure 2) has no close form solution. Indeed, no close form representation is known for the integrals of C s (s) in Equation (2.5). The curve C s (s) in Equation (2.5) is known as the Cornu Spiral [19] curve and its shape is of a spiral of infinitely decreasing radius (increasing curvature). 2.5 Computational Aspect of General Curvature Signature, κ(s) At this point it is obvious that no analytical solution to Equations (2.3) exists for a general κ(s) function. Assume κ(s) is represented as a piecewise polynomial form of degree d κ. piecewise polynomial function of one degree higher [4], d κ + 1. Then, θ(s) = s 0 κ( s)d s is a One can clearly represent the unit circle as a rational form. Yet, herein we are required to employ an arc-length parameterization of the unit circle or Circ(s) = (cos(s), sin(s)). Clearly, Circ(s) is not rational. Let Circ(s) be a polynomial approximation of degree d c to Circ(s), Then, Circ(θ(s)) is a piecewise polynomial vector function that could be evaluated as a composition [8] operation between piecewise polynomial functions. Circ(θ(t)) is of degree d c (d κ + 1). Finally, s 0 Circ(θ( s))d s is a piecewise polynomial vector function of one degree higher, d c (d κ + 1) + 1. We exploited the functionality of the IRIT [12] solid modeler to carry out all the necessary symbolic computations following [7], resulting in curve C(s) being a B-spline curve approximating the reconstructed shape up to the accuracy of the arc-length approximation of Circ(s). To sum up, in Section 2.1, we have presented a tractable scheme to derive the curvature signature, κ(s), of a given curve C(t). Further, in Section 2.2, we have shown how to reconstruct C(t) from κ(s). In Section 3, we will employ these tools to perform an appealing metamorphosis blend between two parametric planar curves. 3 Metamorphosis Using Curvature Interpolation The morphing problem can be defined as the continuous deformation M: M(t, α(s α ), β(s β )), t [0, 1], such that M(0, α(s α ), β(s β )) = α(s α ), 6

7 M(1, α(s α ), β(s β )) = β(s β ), where the first parameter t coincides with the time for fixing the intermediate curve, and α(s α ) and β(s β ) are the two given key curves, with arc-length parameterization s α and s β, respectively. In this work, we present an algorithm to compute the metamorphosis of two-dimensional free-form parametric curves that exploits intrinsic shape parameters. That is, the shape of the intermediate curves, denoted by M(t, α(s α ), β(s β )), depends on the intrinsic geometric properties of the two given curves α(s α ) and β(s β ). Let r [0, 1] and denote by L α and L β the lengths of curves α( ) and β( ), respectively. Further, let s α = rl α and s β = rl β be the arc-length parameters of the two curves that correspond to the former value r. Then, the idea of the presented approach is to linearly interpolate the curvature signatures of the curve M(t, α(s α ), β(s β )) from the two curvature signatures of α(s α ) and β(s β ), κ α (s α ) and κ β (s β ): κ t (s t ) = κ t (rl t ) = (1 t)κ α (rl α ) + tκ β (rl β ) = (1 t)κ α (s α ) + tκ β (s β ), (3.1) where L t = (1 t)l α + tl β is the length of the interpolated curve M (t, α(s α ), β(s β )) at time t. Note the length L t of the intermediate curve is a monotone change of length function between length L α and length L β. Once the curvature, κ t (s t ), of the intermediate planar curve is known, we can recover the curve, following Equation (2.3). One may also apply rigid motion transformation to the interpolated result in order to set the curve s orientation and position properly (recall that the curvature is rotation and translation invariant). 3.1 Piecewise Interpolation We should remember that, thus far, the above approach has not considered the correspondence between the parameterizations of the curves α( ) and β( ). { } N Assume that we are given a set of N matched points, s α i, sβ i, along the two key curves, α(s) i=1 and β(s). Subdivide curves α(s) and β(s) at parameter values s α i and s β i, creating N + 1 pairs of curve segments (assuming α(s) and β(s) are open curves), and then use the curvature interpolation scheme for each pair of segments. The reconstruction of the whole intermediate curve from its segments necessitates the rotation and translation of each segment so as to match the end position and tangent of the previous curve segment, achieving G 2 continuity, as the curvature is matched by construction. An important feature of this reconstruction scheme is identity preserving. That is, if the matched { } values are proportional in each pair s α i, sβ i, i, or s α i = r i L α and s β i = r i L β, for all i {1,..., N}, then 7

8 the metamorphosis sequence is identical to the one that is received via a simple curvature interpolation of the curves, as a whole. 3.2 Metamorphosis of Closed Curves While interpolating between two, beginning and end, closed curves, the intermediate reconstructed curve C t (s) may end up open, i.e. C t (0) C t (L t ). The method to overcoming this difficulty, used in [14], is employed herein as well. Nonetheless, instead of the polygon s vertices we move the control points of the control polygon of C t (s). Let P 0 and P n be the two end points of the control polygon of C t (s). We distribute the error e = P n P 0 evenly, between the points of the control polygon of the curve: where 0 i n. Now, the new control polygon P i = P i i n e, { P i } n i=0 is closed, having P 0 = P n. Nevertheless, tangent continuity is still not guaranteed. The following conditions must be satisfied for C 1 continuity: P n = P 0 and P 0 P 1 = γ (P n P n 1 ), γ > 0. C 1 continuity could be accomplished as follows. Let v = P n 1 (2P 0 P 1 ). Then, assuming P n = P 0, P i = P i i 1 v, where 1 i n 1. n 2 This process affects the G 2 continuity at the end. A similar propagation scheme of the end condition error can be defined using P 0, P 1, P 2 and P n, P n 1, P n 2 to update P n 2 and regain curvature continuity. 4 Examples We now consider some illustrative examples of metamorphosis sequences that have been created using our scheme. The first example (see Figures 3 and 4) presents metamorphosis sequences between a Cornu spiral with curvature κ α (s) = s (see Equation (2.4)) and a Cornu spiral with curvature κ β (s) = 0.5s. The sequence in Figure 3 is generated by our proposed algorithm. The curve is reconstructed as described in Section 2. Compare this metamorphosis sequence with Figure 4 where a regular linear interpolation between the control points of the spirals has been performed. The regular linear interpolation results in self-intersections of the intermediate curves and unnatural looking deformations. The respective curvature signatures of the metamorphosis sequences of Figures 3 and 4 are presented in Figures 5 and 6. Note the extreme curvature values that are reached when linear interpolation is used even for this very simple case. 8

9 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Figure 3: A metamorphosis sequence between two spirals with different linear curvature functions and signs, using curvature interpolation. Compare with Figure 4. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Figure 4: A metamorphosis sequence between two spirals by linear interpolation. Compare with Figure 3. The next example (Figures 7 and 8) represents a metamorphosis sequence between the letters U (1) and S (10). As in the previous example, Figure 7 is the result of using our method, while Figure 8 has been created using linear interpolation between the two symbols. Clearly, Figure 7 looks far more natural. A more interesting example of metamorphosis (between the two human shapes) is presented in Figures 9, 10 and 11. The end user (the artist designing the animation) can anticipate how the intermediate curves are to behave. Such a natural animation is produced by our algorithm (Figure 10), whereas the trivial linear interpolation (Figure 11) results in self-intersection in the arm region and distortions of the arms and legs of the human shapes. A global curvature interpolation has been used in Figure 9, while in Figure 10 9

10 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Figure 5: Curvature signature κ t (s) for the metamorphosis sequence between the two spirals (see Figure 3) with different curvature signs and different curvature change rates, using curvature interpolation. Compare with Figure 6. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Figure 6: Curvature signature κ t (s) for the metamorphosis sequence between the two spirals (see Figure 4) by linear interpolation. Note the extreme curvature values and compare with Figure 5. we subdivided the periodic curves into nine matched segments and performed the metamorphosis in a piecewise manner (see Section 3.1). The nine matched pairs of points have been specified manually for this specific example, whereas, for example, an automatic method that finds curvature extremum points might also be used. 10

11 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Figure 7: Metamorphosis sequence between U (1) and S (10) shapes using curvature interpolation. The length of the intermediate curves is changing monotonically. Compare with Figure 8. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Figure 8: Metamorphosis sequence between U (1) and S (10) shapes using linear interpolation. Compare with Figure 7. While the curvature interpolation scheme produces appealing results, it is also local and hence can easily handle non-simple curves as in Figure 9 (10), having self-intersecting curves as input. Our final example consists of elephant shapes (see Figures 12 and 13). Again, the metamorphosis sequence in Figure 12 produced by our algorithm looks more appealing than the linear interpolation in Figure 13. The tail and the trunk should not shrink or grow wider during the animation (compare Figures 12 11

12 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Figure 9: Metamorphosis sequence between two human shapes (1) and (12) using curvature interpolation. Note the blending of the arms. Compare with Figure 10 and Figure 11. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Figure 10: Metamorphosis sequence between two human shapes (1) and (12) using curvature interpolation and curve segmentation at the points that are labeled by circles. Compare with Figure 9 and Figure 11. and 13). Both given curves have been subdivided into eleven segments and piecewise metamorphosis has been executed. 5 Conclusion In this work, a new method to compute the metamorphosis of two-dimensional parametric curves has been introduced. The solution to the metamorphosis problem requires the solution of two independent tasks. 12

13 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Figure 11: Metamorphosis sequence between two human shapes (1) and (12) by linear interpolation. Compare with Figure 9 and Figure 10. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Figure 12: Metamorphosis sequence between two elephant shapes (1) and (12) using curvature interpolation and curve segmentation at the points that are labeled by circles. Compare with Figure 13. The first derives a correspondence between the two objects. Our focus has been on the second subproblem that deals with a proper, continuous, appealing interpolation between the two shapes. The proposed method interpolates linearly between the curvature signature functions of the two given key curves. When the sequence of the curvature signature functions is known, we are able to reconstruct the metamorphosis sequence itself. The resulting animation sequences have a natural appearance without artificial shrinkage and distor- 13

14 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Figure 13: Metamorphosis sequence between two elephant shapes (1) and (12) by linear interpolation. Compare with Figure 12. tions. The curvature signature can also provide us with global properties such as the winding number of the curve [3]. Hence, a metamorphosis of the two simple curves will neither introduce a double loop curve nor a self-intersecting eight shape curve. Regrettably, this new method does not guarantee self-intersection free intermediate curves even when the given two key curves are known to be simple. This remains an open question. Finally, the extension of this work to free form surfaces is an obvious goal of high value and merit. 6 Acknowledgment The authors would like to thank Alex Sherman for his reference to the Cornu Spiral. This research was supported in part by the Fund for Promotion of Research at the Technion, IIT, Haifa, Israel. References [1] M. Alexa, D. Cohen-Or, and D. Levin, As-rigid-as-possible shape interpolation, Computer Graphics (Proceedings of SIGGRAPH 00), (July 2000), pp [2] N. Arad and D. Reisfeld, Image warping using few anchor points and radial functions, Computer Graphics Forum, 14 (1995), pp

15 [3] M. D. Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, [4] E. Cohen, R. F. Riesenfeld, and G. Elber, Geometric Modeling with Splines - An Introduction, A. K. Peters. Natick, Massachusetts, [5] S. Cohen, G. Elber, and R. Bar-Yehuda, Matching of freeform curves, CAD, 29 (1997), pp Also Center for Intelligent Systems Tech. Report, CIS 9527, Computer Science Department, Technion. [6] D. Cohen-Or, D. Levin, and A. Solomovici, Three-dimensional distance field metamorphosis, ACM Transaction on Graphics, 17 (1998). [7] G. Elber, Free form surface analysis using a hybrid of symbolic and numeric computation, PhD thesis, Computer Science Dept., University of Utah, [8] G. Elber, Symbolic and numeric computation in curve interrogation, Computer Graphics Forum, 14 (1995), pp [9] P. Gao and T. W. Sederberg, A work minimization approach to image morphing, The Visual Computer, 14 (1998), pp [10] E. Goldstein and C. Gotsman, Polygon morphing using a multiresolution representation, Proceeding of Graphics Interface, (1995). [11] J. Hughes, Scheduled Fourier volume morphing, Computer Graphics (SIGGRAPH 92), 26 (1992), pp [12] IRIT, Irit 7.0 User s Manual., Technion, Israel. irit/, Mar [13] T. Samoilov and G. Elber, Self-intersection elimination in metamorphosis of two-dimensional curves, The Visual Computer, 14 (1998), pp [14] T. W. Sederberg, P. Gao, G. Wang, and H. Mu, 2D shape blending: an intrinsic solution to the vertex path problem, Computer Graphics (SIGGRAPH 93), 27 (1993), pp [15] T. W. Sederberg and E. Greenwood, A physically based approach to 2D shape blending, Computer Graphics (SIGGRAPH 92), 26 (1992), pp [16] T. W. Sederberg and E. Greenwood, Shape blending of 2-d piecewise curves, Mathematical Methods in CAGD III, 26 (1995), pp

16 [17] M. Shapira and A. Rappoport, Shape blending using the star-skeleton representation, IEEE Trans. on Computer Graphics and Application, 15 (1995), pp [18] V. Surazhsky and C. Gotsman, Guaranteed intersection-free polygon morphing, Computers and Graphics, (2001), pp [19] E. Weisstein, World of Mathematics, A Wolfram Web Resource, [20] G. Wolberg, Image morphing: a survey, The Visual Computer, 14 (1998), pp

### Intrinsic Morphing of Compatible Triangulations. VITALY SURAZHSKY CRAIG GOTSMAN

International Journal of Shape Modeling Vol. 9, No. 2 (2003) 191 201 c World Scientific Publishing Company Intrinsic Morphing of Compatible Triangulations VITALY SURAZHSKY vitus@cs.technion.ac.il CRAIG

### Research Article Polygon Morphing and Its Application in Orebody Modeling

Mathematical Problems in Engineering Volume 212, Article ID 732365, 9 pages doi:1.1155/212/732365 Research Article Polygon Morphing and Its Application in Orebody Modeling Hacer İlhan and Haşmet Gürçay

### Tatiana Surazhsky Applied Mathematics Department, The Technion Israel Institute of Technology, Haifa 32000, Israel.

Tatiana Surazhsky Applied Mathematics Department, The Technion Israel Institute of Technology, Haifa 32000, Israel. tess@cs.technion.ac.il Gershon Elber Faculty of Computer Science, The Technion IIT, Haifa

### Tatiana Surazhsky Applied Mathematics Department, The Technion Israel Institute of Technology, Haifa 32000, Israel.

Tatiana Surazhsky Applied Mathematics Department, The Technion Israel Institute of Technology, Haifa 32000, Israel. tess@cs.technion.ac.il Gershon Elber Faculty of Computer Science, The Technion IIT, Haifa

### Warping and Morphing. Ligang Liu Graphics&Geometric Computing Lab USTC

Warping and Morphing Ligang Liu Graphics&Geometric Computing Lab USTC http://staff.ustc.edu.cn/~lgliu Metamorphosis "transformation of a shape and its visual attributes" Intrinsic in our environment Deformations

### Polynomial Decomposition and Its Applications

Polynomial Decomposition and Its Applications Joon-Kyung Seong School of Computer Science and Engineering Seoul National University, Seoul 151-742, South Korea E-mail: swallow@3map.snu.ac.kr and Gershon

### Generalized Filleting and Blending Operations toward Functional and Decorative Applications

Generalized Filleting and Blending Operations toward Functional and Decorative Applications Gershon Elber Computer Science Department Technion Haifa 3000, Israel Email: gershon@cs.technion.ac.il Abstract

### Morphing Planar Graphs in Spherical Space

Morphing Planar Graphs in Spherical Space Stephen G. Kobourov and Matthew Landis Department of Computer Science University of Arizona {kobourov,mlandis}@cs.arizona.edu Abstract. We consider the problem

### Blending polygonal shapes with di!erent topologies

Computers & Graphics 25 (2001) 29}39 Shape Blending Blending polygonal shapes with di!erent topologies Tatiana Surazhsky, Vitaly Surazhsky, Gill Barequet, Ayellet Tal * Department of Applied Mathematics,

### Shape Blending Using the Star-Skeleton Representation

Shape Blending Using the Star-Skeleton Representation Michal Shapira Ari Rappoport Institute of Computer Science, The Hebrew University of Jerusalem Jerusalem 91904, Israel. arir@cs.huji.ac.il Abstract:

### 05 - Surfaces. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Daniele Panozzo

05 - Surfaces Acknowledgements: Olga Sorkine-Hornung Reminder Curves Turning Number Theorem Continuous world Discrete world k: Curvature is scale dependent is scale-independent Discrete Curvature Integrated

### Interactively Morphing Irregularly Shaped Images Employing Subdivision Techniques

Interactively Morphing Irregularly Shaped Images Employing Subdivision Techniques Jan Van den Bergh Fabian Di Fiore Johan Claes Frank Van Reeth Expertise Centre for Digital Media - Limburg University Centre

### (Discrete) Differential Geometry

(Discrete) Differential Geometry Motivation Understand the structure of the surface Properties: smoothness, curviness, important directions How to modify the surface to change these properties What properties

### Reparametrization of Interval Curves on Rectangular Domain

International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:15 No:05 1 Reparametrization of Interval Curves on Rectangular Domain O. Ismail, Senior Member, IEEE Abstract The

### COMPUTER AIDED ENGINEERING DESIGN (BFF2612)

COMPUTER AIDED ENGINEERING DESIGN (BFF2612) BASIC MATHEMATICAL CONCEPTS IN CAED by Dr. Mohd Nizar Mhd Razali Faculty of Manufacturing Engineering mnizar@ump.edu.my COORDINATE SYSTEM y+ y+ z+ z+ x+ RIGHT

### Zulfiqar Habib and Manabu Sakai. Received September 9, 2003

Scientiae Mathematicae Japonicae Online, e-004, 63 73 63 G PH QUINTIC SPIRAL TRANSITION CURVES AND THEIR APPLICATIONS Zulfiqar Habib and Manabu Sakai Received September 9, 003 Abstract. A method for family

### Approximation of 3D-Parametric Functions by Bicubic B-spline Functions

International Journal of Mathematical Modelling & Computations Vol. 02, No. 03, 2012, 211-220 Approximation of 3D-Parametric Functions by Bicubic B-spline Functions M. Amirfakhrian a, a Department of Mathematics,

### Shape Modeling. Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell. CS 523: Computer Graphics, Spring 2011

CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell 2/15/2011 1 Motivation Geometry processing: understand geometric characteristics,

### The Convex Hull of Rational Plane Curves

Graphical Models 63, 151 162 (2001) doi:10.1006/gmod.2001.0546, available online at http://www.idealibrary.com on The Convex Hull of Rational Plane Curves Gershon Elber Department of Computer Science,

### Geometric approximation of curves and singularities of secant maps Ghosh, Sunayana

University of Groningen Geometric approximation of curves and singularities of secant maps Ghosh, Sunayana IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish

### Space Curves of Constant Curvature *

Space Curves of Constant Curvature * 2-11 Torus Knot of constant curvature. See also: About Spherical Curves Definition via Differential Equations. Space Curves that 3DXM can exhibit are mostly given in

### Dgp _ lecture 2. Curves

Dgp _ lecture 2 Curves Questions? This lecture will be asking questions about curves, their Relationship to surfaces, and how they are used and controlled. Topics of discussion will be: Free form Curves

### Tutorial 4. Differential Geometry I - Curves

23686 Numerical Geometry of Images Tutorial 4 Differential Geometry I - Curves Anastasia Dubrovina c 22 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves Differential Geometry

### Lecture 11 Differentiable Parametric Curves

Lecture 11 Differentiable Parametric Curves 11.1 Definitions and Examples. 11.1.1 Definition. A differentiable parametric curve in R n of class C k (k 1) is a C k map t α(t) = (α 1 (t),..., α n (t)) of

### Morphing Planar Graphs in Spherical Space

Morphing Planar Graphs in Spherical Space Stephen G. Kobourov and Matthew Landis Department of Computer Science University of Arizona {kobourov,mlandis}@cs.arizona.edu Abstract. We consider the problem

### Central issues in modelling

Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction includes: manual modelling; fitting to

### Almost Curvature Continuous Fitting of B-Spline Surfaces

Journal for Geometry and Graphics Volume 2 (1998), No. 1, 33 43 Almost Curvature Continuous Fitting of B-Spline Surfaces Márta Szilvási-Nagy Department of Geometry, Mathematical Institute, Technical University

### Shape Modeling and Geometry Processing

252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry

### SPIRAL TRANSITION CURVES AND THEIR APPLICATIONS. Zulfiqar Habib and Manabu Sakai. Received August 21, 2003

Scientiae Mathematicae Japonicae Online, e-2004, 25 262 25 SPIRAL TRANSITION CURVES AND THEIR APPLICATIONS Zulfiqar Habib and Manabu Sakai Received August 2, 200 Abstract. A method for family of G 2 planar

### Texture Mapping using Surface Flattening via Multi-Dimensional Scaling

Texture Mapping using Surface Flattening via Multi-Dimensional Scaling Gil Zigelman Ron Kimmel Department of Computer Science, Technion, Haifa 32000, Israel and Nahum Kiryati Department of Electrical Engineering

### Animation. CS 4620 Lecture 32. Cornell CS4620 Fall Kavita Bala

Animation CS 4620 Lecture 32 Cornell CS4620 Fall 2015 1 What is animation? Modeling = specifying shape using all the tools we ve seen: hierarchies, meshes, curved surfaces Animation = specifying shape

### Space Deformation using Ray Deflectors

Space Deformation using Ray Deflectors Yair Kurzion and Roni Yagel Department of Computer and Information Science The Ohio State University Abstract. In this paper we introduce a new approach to the deformation

### Computer Animation Fundamentals. Animation Methods Keyframing Interpolation Kinematics Inverse Kinematics

Computer Animation Fundamentals Animation Methods Keyframing Interpolation Kinematics Inverse Kinematics Lecture 21 6.837 Fall 2001 Conventional Animation Draw each frame of the animation great control

### Physically-Based Modeling and Animation. University of Missouri at Columbia

Overview of Geometric Modeling Overview 3D Shape Primitives: Points Vertices. Curves Lines, polylines, curves. Surfaces Triangle meshes, splines, subdivision surfaces, implicit surfaces, particles. Solids

### A second order algorithm for orthogonal projection onto curves and surfaces

A second order algorithm for orthogonal projection onto curves and surfaces Shi-min Hu and Johannes Wallner Dept. of Computer Science and Technology, Tsinghua University, Beijing, China shimin@tsinghua.edu.cn;

### Parameterization of triangular meshes

Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to

### Circular Arc Approximation by Quartic H-Bézier Curve

Circular Arc Approximation by Quartic H-Bézier Curve Maria Hussain Department of Mathematics Lahore College for Women University, Pakistan maria.hussain@lcwu.edu.pk W. E. Ong School of Mathematical Sciences

### Differential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder]

Differential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder] Preliminaries Recall: Given a smooth function f:r R, the function

### Parametric curves. Reading. Curves before computers. Mathematical curve representation. CSE 457 Winter Required:

Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Parametric curves CSE 457 Winter 2014 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics and Geometric

### Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur

Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur Email: jrkumar@iitk.ac.in Curve representation 1. Wireframe models There are three types

### Refolding Planar Polygons

Refolding Planar Polygons Hayley N. Iben James F. O Brien Erik D. Demaine University of California, Berkeley Massachusetts Institute of Technology Abstract This paper describes a guaranteed technique for

### Animation. CS 4620 Lecture 33. Cornell CS4620 Fall Kavita Bala

Animation CS 4620 Lecture 33 Cornell CS4620 Fall 2015 1 Announcements Grading A5 (and A6) on Monday after TG 4621: one-on-one sessions with TA this Friday w/ prior instructor Steve Marschner 2 Quaternions

### Freeform Curves on Spheres of Arbitrary Dimension

Freeform Curves on Spheres of Arbitrary Dimension Scott Schaefer and Ron Goldman Rice University 6100 Main St. Houston, TX 77005 sschaefe@rice.edu and rng@rice.edu Abstract Recursive evaluation procedures

### Polynomial/Rational Approximation of Minkowski Sum Boundary Curves 1

GRAPHICAL MODELS AND IMAGE PROCESSING Vol. 60, No. 2, March, pp. 136 165, 1998 ARTICLE NO. IP970464 Polynomial/Rational Approximation of Minkowski Sum Boundary Curves 1 In-Kwon Lee and Myung-Soo Kim Department

### The correspondence problem. A classic problem. A classic problem. Deformation-Drive Shape Correspondence. Fundamental to geometry processing

The correspondence problem Deformation-Drive Shape Correspondence Hao (Richard) Zhang 1, Alla Sheffer 2, Daniel Cohen-Or 3, Qingnan Zhou 2, Oliver van Kaick 1, and Andrea Tagliasacchi 1 July 3, 2008 1

### Comparison and affine combination of generalized barycentric coordinates for convex polygons

Annales Mathematicae et Informaticae 47 (2017) pp. 185 200 http://ami.uni-eszterhazy.hu Comparison and affine combination of generalized barycentric coordinates for convex polygons Ákos Tóth Department

### Technion - Computer Science Department - Technical Report CIS

Line Illustrations 2 Computer Graphics Gershon Elber Department of Computer Science Technion, Israel Institute of Technology Haifa 32000 Israel Email: gershon@cs.technion.ac.il January 9, 1994 Abstract

### Parameterization. Michael S. Floater. November 10, 2011

Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point

### Multi-Scale Free-Form Surface Description

Multi-Scale Free-Form Surface Description Farzin Mokhtarian, Nasser Khalili and Peter Yuen Centre for Vision Speech and Signal Processing Dept. of Electronic and Electrical Engineering University of Surrey,

### CS 523: Computer Graphics, Spring Shape Modeling. Skeletal deformation. Andrew Nealen, Rutgers, /12/2011 1

CS 523: Computer Graphics, Spring 2011 Shape Modeling Skeletal deformation 4/12/2011 1 Believable character animation Computers games and movies Skeleton: intuitive, low-dimensional subspace Clip courtesy

### 3D Geometric Metamorphosis based on Harmonic Map

3D Geometric Metamorphosis based on Harmonic Map Takashi KANAI Hiromasa SUZUKI Fumihiko KIMURA Department of Precision Machinery Engineering The University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo 113,

### Digital Geometry Processing Parameterization I

Problem Definition Given a surface (mesh) S in R 3 and a domain find a bective F: S Typical Domains Cutting to a Disk disk = genus zero + boundary sphere = closed genus zero Creates artificial boundary

### Knot Insertion and Reparametrization of Interval B-spline Curves

International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:14 No:05 1 Knot Insertion and Reparametrization of Interval B-spline Curves O. Ismail, Senior Member, IEEE Abstract

### A Hole-Filling Algorithm for Triangular Meshes. Abstract

A Hole-Filling Algorithm for Triangular Meshes Lavanya Sita Tekumalla, Elaine Cohen UUCS-04-019 School of Computing University of Utah Salt Lake City, UT 84112 USA December 20, 2004 Abstract Data obtained

### Differential Geometry MAT WEEKLY PROGRAM 1-2

15.09.014 WEEKLY PROGRAM 1- The first week, we will talk about the contents of this course and mentioned the theory of curves and surfaces by giving the relation and differences between them with aid of

### Computer Graphics Curves and Surfaces. Matthias Teschner

Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves

### 2D Spline Curves. CS 4620 Lecture 18

2D Spline Curves CS 4620 Lecture 18 2014 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes that is, without discontinuities So far we can make things with corners (lines,

### Background for Surface Integration

Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to

### CS 523: Computer Graphics, Spring Differential Geometry of Surfaces

CS 523: Computer Graphics, Spring 2009 Shape Modeling Differential Geometry of Surfaces Andrew Nealen, Rutgers, 2009 3/4/2009 Recap Differential Geometry of Curves Andrew Nealen, Rutgers, 2009 3/4/2009

### Until now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple

Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of

### 3D Modeling Parametric Curves & Surfaces

3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision

### View Interpolation for Dynamic Scenes

EUROGRAPHICS 2002 / I. Navazo Alvaro and Ph. Slusallek (Guest Editors) Short Presentations View Interpolation for Dynamic Scenes Jiangjian Xiao Cen Rao Mubarak Shah Computer Vision Lab School of Electrical

### Singularity Loci of Planar Parallel Manipulators with Revolute Joints

Singularity Loci of Planar Parallel Manipulators with Revolute Joints ILIAN A. BONEV AND CLÉMENT M. GOSSELIN Département de Génie Mécanique Université Laval Québec, Québec, Canada, G1K 7P4 Tel: (418) 656-3474,

### Geometry and Gravitation

Chapter 15 Geometry and Gravitation 15.1 Introduction to Geometry Geometry is one of the oldest branches of mathematics, competing with number theory for historical primacy. Like all good science, its

### Kinematics of Closed Chains

Chapter 7 Kinematics of Closed Chains Any kinematic chain that contains one or more loops is called a closed chain. Several examples of closed chains were encountered in Chapter 2, from the planar four-bar

### Curve and Surface Basics

Curve and Surface Basics Implicit and parametric forms Power basis form Bezier curves Rational Bezier Curves Tensor Product Surfaces ME525x NURBS Curve and Surface Modeling Page 1 Implicit and Parametric

### CS337 INTRODUCTION TO COMPUTER GRAPHICS. Describing Shapes. Constructing Objects in Computer Graphics. Bin Sheng Representing Shape 9/20/16 1/15

Describing Shapes Constructing Objects in Computer Graphics 1/15 2D Object Definition (1/3) Lines and polylines: Polylines: lines drawn between ordered points A closed polyline is a polygon, a simple polygon

### Generalized barycentric coordinates

Generalized barycentric coordinates Michael S. Floater August 20, 2012 In this lecture, we review the definitions and properties of barycentric coordinates on triangles, and study generalizations to convex,

### CSE452 Computer Graphics

CSE452 Computer Graphics Lecture 19: From Morphing To Animation Capturing and Animating Skin Deformation in Human Motion, Park and Hodgins, SIGGRAPH 2006 CSE452 Lecture 19: From Morphing to Animation 1

### Three-Dimensional Distance Field Metamorphosis

Three-Dimensional Distance Field Metamorphosis DANIEL COHEN-OR DAVID LEVIN and AMIRA SOLOMOVICI Tel-Aviv University Given two or more objects of general topology, intermediate objects are constructed by

### 04 - Normal Estimation, Curves

04 - Normal Estimation, Curves Acknowledgements: Olga Sorkine-Hornung Normal Estimation Implicit Surface Reconstruction Implicit function from point clouds Need consistently oriented normals < 0 0 > 0

### Animations. Hakan Bilen University of Edinburgh. Computer Graphics Fall Some slides are courtesy of Steve Marschner and Kavita Bala

Animations Hakan Bilen University of Edinburgh Computer Graphics Fall 2017 Some slides are courtesy of Steve Marschner and Kavita Bala Animation Artistic process What are animators trying to do? What tools

### Free-Form Deformation and Other Deformation Techniques

Free-Form Deformation and Other Deformation Techniques Deformation Deformation Basic Definition Deformation: A transformation/mapping of the positions of every particle in the original object to those

### MOTION OF A LINE SEGMENT WHOSE ENDPOINT PATHS HAVE EQUAL ARC LENGTH. Anton GFRERRER 1 1 University of Technology, Graz, Austria

MOTION OF A LINE SEGMENT WHOSE ENDPOINT PATHS HAVE EQUAL ARC LENGTH Anton GFRERRER 1 1 University of Technology, Graz, Austria Abstract. The following geometric problem originating from an engineering

### CS 450 Numerical Analysis. Chapter 7: Interpolation

Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80

### Polyhedron Realization for Shape Transformation

Polyhedron Realization for Shape Transformation Avner Shapiro Ayellet Tal Department of Computer Science Department of Electrical Engineering The Hebrew University Technion Israel Institute of Technology

### CHAPTER 6 Parametric Spline Curves

CHAPTER 6 Parametric Spline Curves When we introduced splines in Chapter 1 we focused on spline curves, or more precisely, vector valued spline functions. In Chapters 2 and 4 we then established the basic

### CGT 581 G Geometric Modeling Curves

CGT 581 G Geometric Modeling Curves Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics Technology Curves What is a curve? Mathematical definition 1) The continuous image of an interval

### An introduction to interpolation and splines

An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve

### Mesh Morphing. Ligang Liu Graphics&Geometric Computing Lab USTC

Mesh Morphing Ligang Liu Graphics&Geometric Computing Lab USTC http://staff.ustc.edu.cn/~lgliu Morphing Given two objects produce sequence of intermediate objects that gradually evolve from one object

### 3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013

3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels

### Construction and smoothing of triangular Coons patches with geodesic boundary curves

Construction and smoothing of triangular Coons patches with geodesic boundary curves R. T. Farouki, (b) N. Szafran, (a) L. Biard (a) (a) Laboratoire Jean Kuntzmann, Université Joseph Fourier Grenoble,

### A Comparison of Gaussian and Mean Curvatures Estimation Methods on Triangular Meshes

A Comparison of Gaussian and Mean Curvatures Estimation Methods on Triangular Meshes Tatiana Surazhsky, Evgeny Magid, Octavian Soldea, Gershon Elber and Ehud Rivlin Center for Graphics and Geometric Computing,

### CS123 INTRODUCTION TO COMPUTER GRAPHICS. Describing Shapes. Constructing Objects in Computer Graphics 1/15

Describing Shapes Constructing Objects in Computer Graphics 1/15 2D Object Definition (1/3) Lines and polylines: Polylines: lines drawn between ordered points A closed polyline is a polygon, a simple polygon

### City Research Online. Permanent City Research Online URL:

Slabaugh, G.G., Unal, G.B., Fang, T., Rossignac, J. & Whited, B. Variational Skinning of an Ordered Set of Discrete D Balls. Lecture Notes in Computer Science, 4975(008), pp. 450-461. doi: 10.1007/978-3-540-7946-8_34

### Motivation. Parametric Curves (later Surfaces) Outline. Tangents, Normals, Binormals. Arclength. Advanced Computer Graphics (Fall 2010)

Advanced Computer Graphics (Fall 2010) CS 283, Lecture 19: Basic Geometric Concepts and Rotations Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283/fa10 Motivation Moving from rendering to simulation,

### Skeletal deformation

CS 523: Computer Graphics, Spring 2009 Shape Modeling Skeletal deformation 4/22/2009 1 Believable character animation Computers games and movies Skeleton: intuitive, low dimensional subspace Clip courtesy

### Normals of subdivision surfaces and their control polyhedra

Computer Aided Geometric Design 24 (27 112 116 www.elsevier.com/locate/cagd Normals of subdivision surfaces and their control polyhedra I. Ginkel a,j.peters b,,g.umlauf a a University of Kaiserslautern,

### CURVES OF CONSTANT WIDTH AND THEIR SHADOWS. Have you ever wondered why a manhole cover is in the shape of a circle? This

CURVES OF CONSTANT WIDTH AND THEIR SHADOWS LUCIE PACIOTTI Abstract. In this paper we will investigate curves of constant width and the shadows that they cast. We will compute shadow functions for the circle,

### ME 115(b): Final Exam, Spring

ME 115(b): Final Exam, Spring 2011-12 Instructions 1. Limit your total time to 5 hours. That is, it is okay to take a break in the middle of the exam if you need to ask me a question, or go to dinner,

### Perceptually Based Approach for Planar Shape Morphing

Perceptually Based Approach for Planar Shape Morphing Ligang Liu Guopu Wang 1 Bo Zhang Baining Guo Heung-Yeung Shum Microsoft Research Asia 1 Tsinghua University Abstract This paper presents a novel approach

### For each question, indicate whether the statement is true or false by circling T or F, respectively.

True/False For each question, indicate whether the statement is true or false by circling T or F, respectively. 1. (T/F) Rasterization occurs before vertex transformation in the graphics pipeline. 2. (T/F)

### Parametric curves. Brian Curless CSE 457 Spring 2016

Parametric curves Brian Curless CSE 457 Spring 2016 1 Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics

### Estimating normal vectors and curvatures by centroid weights

Computer Aided Geometric Design 21 (2004) 447 458 www.elsevier.com/locate/cagd Estimating normal vectors and curvatures by centroid weights Sheng-Gwo Chen, Jyh-Yang Wu Department of Mathematics, National

### Lesson 2: Wireframe Creation

Lesson 2: Wireframe Creation In this lesson you will learn how to create wireframes. Lesson Contents: Case Study: Wireframe Creation Design Intent Stages in the Process Reference Geometry Creation 3D Curve

### CS354 Computer Graphics Surface Representation IV. Qixing Huang March 7th 2018

CS354 Computer Graphics Surface Representation IV Qixing Huang March 7th 2018 Today s Topic Subdivision surfaces Implicit surface representation Subdivision Surfaces Building complex models We can extend

### PS Geometric Modeling Homework Assignment Sheet I (Due 20-Oct-2017)

Homework Assignment Sheet I (Due 20-Oct-2017) Assignment 1 Let n N and A be a finite set of cardinality n = A. By definition, a permutation of A is a bijective function from A to A. Prove that there exist

### Justin Solomon MIT, Spring 2017

http://www.alvinomassage.com/images/knot.jpg Justin Solomon MIT, Spring 2017 Some materials from Stanford CS 468, spring 2013 (Butscher & Solomon) What is a curve? A function? Not a curve Jams on accelerator