CGT 581 G Geometric Modeling Curves

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1 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 in an space 2) A continuous map of a 1 space to an space Curves What is a curve? Definition from physics: curve is a trajectory of a moving point This is usually used in CG Curves Any point has two neighbors (no branching) Endpoints have only one neighbor Some curves do not have endpoints Infinite curves Closed curves Periodic curves Space filling curves are 2D. 1

2 Curve representation Explicit: (!) cannot represent closed curves (and other issues) Implicit:, 0 many useful properties usually evaluated numerically, is a scalar function (one number) Curve representation Parametric:,, The curve is a function of one parameter It is a trajectory of a moving point The meaning of the parameter is the time It returns three (3D) or more (nd) values that identify the coordinates of the point in the n dimensional space Curve representation Trajectory (curve) is thus represented as a set of points in space: Parametric Representation Example Sinus curve, sin Circle sin, cos Their evaluation is straightforward How sparse are they? Parabola, 2 2

3 Parametric Representation Example Different density. Take two curves:, 0 and t,0, 0 1 they represent the same shape of the curve, but,0 1 They represent the same curve with different parameterization does not mean midpoint of a curve segment! Parametric Representation The parameter usually , 0, 0 is the starting point of the curve Also denoted by 1 1, 1, 1 the end point of the curve Also denoted by Parametric Equation of a Line Defined by two points and its equation,, ; there are two special cases: Parametric Equation of a Line Can be also thought of as a blending function 1 That is a linear interpolation between and with the parameter 3

4 Parametric Curve Derivative The derivative of a curve w.r.t the parameter,, Parametric Curve Derivative Example: derivatives of the functions from the previous examples: sinus curve, sin 1, cos circle sin, cos cos, sin parabola 2, 2, 1 Parametric Curve Derivative Note: Curve is a set of points Derivative of a curve is a vector. Why? Take the limit process Δ lim Δ Δ Δ Δ Δ Δ Δ Parametric Curve Derivative The derivative is the tangent vector to the function in the given point Tangent is a line passing through the given point in the direction of the tangent vector 4

5 Tangent & Tangent Vector Example Example: Having a curve , Evaluate the tangent in the point 0.5. Solution: 1. Get the point Evaluate derivative 3. Evaluate tangent vector Tangent is: Tangent & Tangent Vector Example ad , ,0 ad , ad , 3/2 ad 4. 1,0 3, 3/2 by the way: 1, P(s) Q(t) Curvature Measures how a curve deviates from being flat. The tangent vector and the second derivative in a point define osculating plane Curvature The osculating circle lies in and touches at The radius of the osculating curve is The curvature of at is Helmut Pottman, Architectural geometry Three consecutive vertices possess circumscribed circle which lies in the osculating plane the osculating circle also 1 5

6 Curvature Note Curvature of an inflexion point is zero ( Parametric curve continuity Motivation: Expression of the entire shape from only one curve is usually impossible Curves are connected at their ends Every part of the curve is called a segment The point that two segments share is a knot (joint) The parametric continuity is denoted by C Parametric curves continuity knot segment Free form curves continuity 0 If two segments meet in a point, they are continuously connected or simply connected Denoted by 0 Two segments 1 and 2 are continuously connected iff (One segment starts where the other ends) 6

7 Free form curves continuity 1 Two segments 1 and 2 are parametrically continuously connected iff 1) Shared knot 2) 1 0 Derivatives in the knot are equal Does 1 imply 0? Free form curves continuity 1 So called 1 parametric continuity requires two segments to be 0 and their derivatives to be equal in the knot Denoted by 1 Two segments have identical tangent in the knot (not only the tangent vectors!) Magnitude and the direction are identical (If we move from the one curve to the other, speed and the direction is unchanged (continuous)) Continuity 0 vs. 1 Continuity 0 vs. Continuity 0 : Shared knot Q 1 (1)=Q 2 (0) Q 1 (1) Continuity 1 : shared knot, shared tangent, and the tangent vectors are nonzero Q 1 (t) Q 1 (t) Q 2 (0) Q 2 (t) Q 1 (1)=Q 2 (0) Q 1 (1)=Q 2 (0) Q 2 (0) 7

8 Continuity 2, 3, Continuity 2, 3, A curve is said to have continuity iff it has continuous derivatives of order 0, 1,, in every point Two curves meet continuously iff they have continuous derivatives of order 0, 1,, n in every point Note: polynomial functions are Continuity Continuity 2, 3,, Sometimes denoted refers to curves that include discontinuities Example: 8

9 Curves continuity (contd.) Example: 1 sin /2, cos / , /2 0 1 Are they parametrically continuous? Curves continuity (contd.) Solution: ad 1) 1 1 sin /2, cos /2 1, , 0 this is OK 2) 1 /2 cos /2, /2sin /2 2 0, /2 1 1 /2 cos /2, /2sin /2 0, /2 that is OK as well Curves continuity (contd.) These two segments form a curve that is parametrically continuous in the knot. Curves continuity (contd.) Example: Let s have curves: 2, 2, 0 1 2, 2, 0 1 Is the curve formed by them (if any) 1? 9

10 Curves continuity (contd.) Solution: 1) 1 2,2 0 2,2 2) 2,2 1,1 The curves definitively meet smoothly, but they are NOT parametrically continuous. Tangent vectors have different magnitudes! 2,2 1, Geometric Continuity 1 Geometric continuity means visual smoothness. The moving point changes the velocity but not the direction. Two curves are geometrically continuous if their derivatives in knot are positively linearly dependent. Geometric Continuity 1 Two segments 1 and 2 are geometrically continuously connected iff Geometric Continuity 1 1) they share a knot 2) , 0 their derivatives in the knot are linearly dependent Does 1 imply 1? 10

11 Geometric Continuity The direction of the second derivative are identical. Geometric Continuity 1) 1 0 they share a knot 2) 1 0, 0 derivatives in the knot are linearly dependent 3) 1 0, 0 accelerations in the knot are linearly dependent exist if the two connecting curves also share center of curvature at the joint.,, summary curves share knot curves also share tangent curves also share a common center of curvature at the joint. Note: geometric continuity exist if the curves can be re parameterized to have. Continuity summary Continuity (zero order continuity): 0 two curves meet in the knot Parametric continuity (first order continuity): 1 the curves meet in knot and their derivatives are equal (the joint is smooth) Geometric continuity: 1 the curves meet in knot and their derivatives are linearly positively dependent (the joint point is visually smooth) 11

12 Readings Architectural Geometry, Pottman et al Interactive Computer Graphics 5th edition, Ed. Angel pp Real Time Rendering 2nd edition, Moller, T.A., Haines, E., Geometric Modeling, 2nd edition, M.E.Mortenson 12

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