Spline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
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1 Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
2 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong wriggling effects. Solution: Use curves consisting of several low-degree segments. Condition: The polynomial segments have to fit together smoothly at the transitions (nodes). Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 2
3 Definition C k -continuity A function f (t) is C k -coninuous, if the function and its first k derivatives are continuous. C k [t 0, t n ] is the class of C k -continuous functions on the interval [t 0, t n ]. Definition Spline Let τ = {t 0,..., t n } a node vector with real-valued nodes t i < t i+1. A function S is called Spline of degree k (of order k + 1) if: 1 S is a polynomial of degree k in every partial interval [t i, t i+1 ] 2 S is C k 1 -continuous on [t 0, t n ]. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 3
4 Remarks 1 The spline S is called interpolating spline, if S(t i ) = p i for a given set of interpolation points p i (ordinates). 2 In general, the interpolating spline is not uniquely defined. Therefore, additional boundary conditions are required for the remaining k 1 degrees of freedom. Natural Splines: cubic splines (k = 3) with natural boundary conditions S (t 0 ) = 0 and S (t n ) = 0 Periodic Splines: identify nodes t 0 and t n with each other, i.e. S(t 0 ) = S(t n ); S (t 0 ) = S (t n ); S (t 0 ) = S (t n ) Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 4
5 Cubic Splines Instead of specifying properties of the interpolating function (e.g. maximal degree), we can also specify the shape (e.g. smooth ). Requirement: tn t 0 g (t) 2 minimal (1) with additional conditions: g(t j ) = p j (j = 0,..., n), g (t 0 ) = p 0 and g (t n ) = p n (2) Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 5
6 Theorem Minimum-Norm Property Among all functions g C 2 [t 0, t n ] which satisfy g(t i ) = y i, the integral t n t 0 g (x) 2 for the interpolating cubic spline function S has the smallest value. Proof:... Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 6
7 Interpolation with Natural Cubic Splines Coefficients a i, b i, c i, d i of a natural cubic spline: S(t) = S i (t) = a i + b i (t t i ) + c i (t t i ) 2 + d i (t t i ) 3 for t [t i, t i+1 ], i = 0,..., n 1. Conditions for polynomials S i : S i (t i ) = p i, i = 0,..., n 1, S i (t i+1 ) = p i+1, i = 0,..., n 1, S i (t i ) = S i 1(t i ), i = 1,..., n 1, S i (t i ) = S i 1(t i ), i = 1,..., n 1. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 7
8 For our coefficients, this implies that: a i = p i, i = 0,..., n 1 and S n 1 (t n ) = p n, a i = a i 1 + b i 1 (t i t i 1 ) + c i 1 (t i t i 1 ) 2 + d i 1 (t i t i 1 ) 3, i = 1,..., n 1 b i = b i 1 + 2c i 1 (t i t i 1 ) + 3d i 1 (t i t i 1 ) 2, i = 1,..., n 1 2c i = 2c i 1 + 6d i 1 (t i t i 1 ), i = 1,..., n 1 Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 8
9 We define i := t i+1 t i and, after some transformations, get: c i 1 ( i 1 ) + c i (2( i 1 + i )) + c i+1 ( i ) = 3 i (p i+1 p i ) 3 i 1 (p i p i 1 ) i = 1,..., n 1 (3) with c n := 0 d i = 1 3 i (c i+1 c i ) i = 0,..., n 1 (4) b i = 1 i (p i+1 p i ) i 3 (c i+1 + 2c i ) i = 0,..., n 1 (5) Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 9
10 a i = p i, i = 0,..., n 1, i.e. the a i are already known. We have to solve a linear system of n 1 equations with n + 1 unknowns (c i, i = 0,..., n). Natural Cubic Splines S (t 0 ) = 0 and S (t n ) = 0 c 0 = c n = 0 Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
11 We have: n + 1 supporting values (nodes) t i with t 0 < t 1 < < t n and function values p 0,..., p n. We want: natural cubic spline S of the form S(t) = S i (t) = a i +b i (t t i )+c i (t t i ) 2 +d i (t t i ) 3 for t [t i, t i+1 ] and i = 0,..., n 1. The coefficients a i, b i, c i, d i are found using (3)-(5) and a i = p i i = 0,..., n 1 c 0 = c n = 0 Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
12 The equations (3) can be written in matrix form Ac = r: A := 2( ) ( ) n n 2 2( n 2 + n 1 ) c := c 1. c n 1 r := 3 p 2 p 1 1 p 1 p 0 0 p n p n 1 n 1. p n 1 p n 2 n 2 Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
13 The matrix A is tridiagonal, symmetric, diagonally dominant, positive-definite, and consists of only positive elements. This implies: A is regular and the linear system of equations has a unique solution. For solving the system, the direkt LU-decomposition for tridiagonal matrices should be used, because the algorithm has the complexity of only O(n). Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
14 Interpolation with Periodic Cubic Splines If we identify t 0 with t n, i.e. p 0 = p n, we get a closed interpolating curve with a C 2 -continuous transition at t 0. In the algorithm for natural cubic splines, we only have to change the matrix A of system (3): A := 2( ) ( ) n n 1 2( n ) Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
15 This matrix is cyclic-tridiagonal, symmetric, diagonally dominant, positive-definite and consists of only positive elements. This implies: A is well-conditioned The system can also be solved in O(n). In comparison to the system (3) for natural cubic splines, the matrix is larger by one row and one column, because we have n instead of n 1 transitions. Due to the cyclic band structure of the matrix, we have additional non-zero elements in the upper right and lower left corners. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
16 Polynomial Averaging Splines We have: Values f (t i ) = f i of a function f C[a, b] with errors (e.g. measurement errors) n + 1 nodes t i with a = t 0 < t 1 < < t n = b The distribution of the f i makes a useful approximation with interpolating splines impossible. We need an error-compensating replacement function, which runs smoothly along the points (t i, f i ). Consider an interpolating spline S through a new set of values g i. The g i have to satisfy that the differences f i g i are positively proportional to the jumps γ i of the third derivative of S at t i. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
17 Definition Polynomial Averaging Spline A polynomial averaging spline of degree three with n + 1 nodes t i and flawed values f i is a function S : [a, b] R with the following properties: S C 2 [a, b] S is a polynomial of degree three in each sub-interval (7) S(t i ) = g i for i = 0,..., n (8) w i (f i g i ) = γ i for i = 0,..., n with weights w i > 0 (9) (6) γ 0 = S 0 (t 0 ) γ n = S n 1(t n ) γ i = S i (t i ) S i 1(t i ) i = 1,..., n 1 Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
18 Choosing large values for the weights w i will result in a curve that is very close to the data values f i, while smaller w i will result in a smoother curve. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
19 The system of equations used by an averaging spline algorithm can be constructed by combining the conditions for interpolating splines with the specific conditions for averaging splines (9). Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
20 Parametric Splines Modelling of plane curves or space curves: vector-valued or parametric splines Definition Parametric Spline Consider an interval [a, b] R and := (t 0,..., t n ), a = t 0 < t 1 < < t n = b. The mapping X : [a, b] R 3 is called parametric spline of degree k (order k + 1), if its component functions x i, i = 1, 2, 3 are of degree k: x i C k 1 [a, b] (i = 1, 2, 3) short: X C k 1 [a, b] Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
21 Definition C k -Transition Two parametric curves X : [t 0, t 1 ] R 3 and Y : [s 0, s 1 ] R 3 with X C m [t 0, t 1 ] and Y C n [s 0, s 1 ] have a C k -transition at their shared point X (t 1 ) = Y (s 0 ) if the following holds: d r dt r X (t 1) = d r ds r Y (s 0) for all r with 1 r k. Discontinuities are also called C 1 -transitions. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
22 Interpolation with Parametric Cubic Splines We have: interpolation points p i := (x i, y i, z i ), i = 0,..., n. We want: interpolating parametric cubic spline S(t). Step 1: Parametrization Specify parameter values (nodes) t i, i = 0,..., n for the interpolation points: S(t i ) = p i. Step 2: Boundary Conditions Specify boundary conditions (natural or periodic). Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
23 Step 3: Spline algorithm Calculate the spline components S x, S y, S z such that S x (t i ) = x i, S y (t i ) = y i, S z (t i ) = z i, i = 0,..., n using the appropriate spline algorithm: S x (t) = S xi (t) = a xi + b xi (t t i ) + c xi (t t i ) 2 + d xi (t t i ) 3 S y (t) = S yi (t) = a yi + b yi (t t i ) + c yi (t t i ) 2 + d yi (t t i ) 3 S z (t) = S zi (t) = a zi + b zi (t t i ) + c zi (t t i ) 2 + d zi (t t i ) 3 t [t i, t i+1 ], i = 0,..., n 1. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
24 For closed curves which are smooth everywhere, it makes sense to use periodic splines. If a curve has one or more cusps (i.e. C 0 -transitions), we can use natural splines with the cusps as start- and endpoints. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
25 Parametrizations The shape, and therefore also the quality, of a curve (or surface) depends strongly on the parametrization. (a) and (b) are two different parametrizations for the same set of interpolation points. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
26 We can demonstrate the impact of the choice of parameters using a kinematic interpretation: Interpret the curve parameter t as a time parameter. It represents the amount of time that a point S would need to traverse the curve. For the following parametrizations, we are interpolating a set of points using a curve: parameter interval [a, b] and n + 1 points to interpolate Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
27 Equidistant Parametrization For each pair of successive interpolations points (p i, p i+1 ), we have the same amount of time for traversal: t = b a n ; t i = a + i t; i = 0,..., n If the distances between the points vary strongly, then a point S traverses the curve with varying speed. That is, if a large distance is followed by a small distance, the speed has to be reduced greatly. This can result in wiggling of the interpolation curve. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
28 Chordal Parametrization Idea: Adapt the parametrization to the structure of the point set. This is achieved by selecting the parameter intervals proportionally to the distances of neighboring interpolation points. We control the overall length using a normalization factor s (example: s is the overall length of the polygon formed by the p i ). t i = t i+1 t i := p i+1 p i. s Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
29 Centripetal Parametrization Idea: The parametrization should minimize the centripetal acceleration (normal acceleration) [Lee 1989]. The normal forces along an arc are proportional to the angular velocity. t i := pi+1 p i s These parametrizations are not affine invariant because length measurements are used. For example, the expression to the right is affine invariant only if the three points lie in a line. p i+1 p i p i+2 p i+1 Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
30 Foley-Parametrization Idea: The parametrization should take into account distances and angle differences at the interpolation points [Foley 1989]. ( t i := d i φ i d i 1 3(d i 1 + d i ) + 2 φ ) i+1 d i+1 3(d i + d i+1 ) with φ i := min(π φ i, π 2 ). Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
31 Nielson-Metric The distance function d used with the Foley-parametrization can be either the euclidean metric or the Nielson-metric, which is affine-invariant: ( ) x 2 y N ( σy := (x, y) with σ x := 1 n σ xy := 1 n σ xy σ xy σ x ) ( x y n (x i x) 2 ; σ y := 1 n i=1 ) x := 1 n y := 1 n n x i ; i=1 n y i ; i=1 n (y i y) 2 ; i=1 n (x i x) (y i y); := σ x σ y σxy 2 i=1 Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
32 The Nielson-metric works as follows: scale the point set in a way that the variance is equal in all directions distances are obtained from the scaled arrangement of the points The metric allows for independence from coordinate systems and scaling. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
33 Example equidistant parametrization centripetal parametrization chordal parametrization Foley-parametrization with Nielson metric Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
34 Parameter Transformations The shape of a curve is strongly influenced by the parametrization of the interpolation points. It is, however, possible to modify the parametrization without altering the shape of the curve. Definition Parameter Transformation, Reparametrization Consider a parametric curve X (t) and a bijective, continuous function ϕ(t). Then the curve Y (t) : X (ϕ(t)) can be obtained from X using a parameter transformation. If both ϕ and ϕ 1 are continuously differentiable, ϕ is called a C 1 parameter transformation. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
35 Example The length of a curve segment is L(t 0, t 1 ) = t1 t 0 Ẋ (t) dt, Ẋ = dx dt Using this, we can, for example, re-parametrize the curve such that Ẋ (t) = 1 and t is the arc length of the curve. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
36 The following properties do not depend on the parametrization of the curve: Curvature ( Deviation with respect to the tangent ) and Torsion ( Deviation from planar shape ) Definition κ(t) = τ(t) = Ẋ (t) Ẍ (t) Ẋ (t) 3 (Ẋ... (t), Ẍ (t), det Ẋ ) X (t) (t) Ẍ (t) 2 (curvature) (torsion) The radius of the osculating circle (German: Schmiegekreis) of X (t) is r(t) = 1 κ(t). Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
37 Example Proof that the curvature of Y (t) = X (ϕ(t)) does not depend on ϕ: Y (t) = dy dt = X (ϕ(t))ϕ (t) =: X ϕ Ÿ (t) = d Y dt = X (ϕ(t))(ϕ (t)) 2 + X (ϕ(t))ϕ (t) =: X ϕ 2 + X ϕ (X ϕ ) (X ϕ 2 + X ϕ ) κ Y (t) = X ϕ 3 = ϕ 3 X X + ϕ ϕ = X X X 3 ϕ 3 X 3 = κ X (ϕ(t)) 0 { }}{ X X Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
38 Remark Instead of C k -continuity, continuity of geometric properties (e.g. tangent, curvature, torsion,...) may be required. As a result, we have more degrees of freedom for modelling. geometric splines, G-splines see lectures on Geometric Modelling Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/
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