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

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1 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 n! different permutations of A. Assignment 2 Prove the following three claims for all n N and all x R: n i=0 ( ) n x i = (1 + x) n i n i=0 ( ) n i n ( ) n = 2 n ( 1) i = 0 i Assignment 3 Prove that the following identity holds for all n, c N 0 : n k=0 ( ) k = c ( ) n + 1 c + 1 Assignment 4 In R[x] we consider the polynomials a, b, c, d with a := 1 + x + x 2 + x 3, b := x + x 2, c := x, d := 3 + 2x + x 3, e := x 4. Do {a, b, c, d, e} form a basis of the vector space of all polynomials of R[x] whose degree is at most four? i=0

2 Homework Assignment Sheet II (Due 10-Nov-2017) Assignment 5 Apply Euklid s Algorithm to compute the greatest common divisor, gcd(p, q), of p, q R[x] with p := x 3 3x 2 + 5x 3 and q := x 3 1. (Note: Euclid s Algorithm for computing the GCD of two integers can also be applied in polynomial rings, and division of two polynomials is similar to what you had learned in school for dividing one integer by another integer by hand.) Assignment 6 Prove that f(x) := cos(2 arccos(x)) is a polynomial function over [ 1, 1]. Assignment 7 Prove that two different polynomials may result in the same polynomial function. Assignment 8 Prove explicitly (based on Def. 19) that f : R R with f(x) := x 2 is differentiable on all of R.

3 Homework Assignment Sheet III (Due 17-Nov-2017) Assignment 9 In the lecture we discussed that the length of the Koch snowflake after the n-th iteration is ( 4 /3) n s if the original triangle had a perimeter of s. Hence, its length grows unboundedly as n increases. What about the area enclosed by the Koch snowflake? Your task is to formulate a formula that models the area enclosed by the Koch snowflake after the n-th iteration. What do you get for the area for n? (You are welcome to assume that the Koch snowflake is a simple curve after every iteration.) Assignment 10 Prove precisely that in the proof of Lem. 49 equality at ( ) does indeed hold. (Of course, you are welcome to assume the correctness of the Mean Value Theorem.) Assignment 11 Find a reparameterization β of γ : R + 0 R 3, with γ(t) := (a cos t, a sin t, bt) for a, b R +, which has unit speed. What is the arc length of β in dependence on the parameter t? Assignment 12 We consider a regular curve γ : I R 2, for some (non-empty) interval I R. Your task is to compute the (algebraic) equation of the tangent in γ(t), for some t I.

4 Homework Assignment Sheet IV (Due 24-Nov-2017) Assignment 13 Find parameterizations of two curves that join in a C 1 - and curvaturecontinuous fashion but that are not C 2 -continuous. Assignment 14 Compute the equation of the plane that is tangential to the graph of the function f(x, y) := e x2 y 2 in the point P := (1, 1, 1/e 2 ). Assignment 15 Let β : [ π, 0] R 3 and γ : [0, π] R 3 be defined as follows: β(t) := ( 1 + cos t, sin t, t) γ(t) := (2 2 cos t, 2 sin t, 2t) (1) Prove that β, γ meet in a G 1 -continuous way but that they are not C 1 -continuous. (2) Find an equivalent reparameterization δ of γ such that β and δ are C 1 -continuous at the joint. Assignment 16 Prove directly without resorting to (the proofs of) Lem. 83, Lem. 84 or Thm. 86! that B 0,2 (x), B 1,2 (x) and B 2,2 (x) are linearly independent.

5 Homework Assignment Sheet V (Due 01-Dec-2017) Assignment 17 Show directly (without resorting to the general theorem on derivatives of Bernstein basis polynomials) that B k,n(x) = n ( B k 1,n 1 (x) B k,n 1 (x) ) for all k, n N 0 with k n. Assignment 18 Prove or disprove: If a control polygon is symmetric about the y-axis then also the Bézier curve defined by this control polygon is symmetric about the y-axis. (You are welcome to consider only Bézier curves of either odd or even degree, as you prefer.) Assignment 19 Show that a line may intersect the control polygon of a Bézier curve without actually intersecting the Bézier curve itself. Assignment 20 Let B be the Bézier curve defined by the four control points p 0 := (0, 0) p 1 := (25, 50) p 2 := (50, 0) p 3 := ( 50, 25). Run de Casteljau s algorithm manually to compute B( 3 /5).

6 Homework Assignment Sheet VI (Due 15-Dec-2017) Assignment 21 Consider the curve α: R R 2 with α(t) := (3t + 3t 2, 1 + 4t 3 ) and compute a Bézier curve B such that B = α [0,1]. Assignment 22 Consider a Bézier curve B n defined by the control points p 0 := ( 2, 0), p 1 = p 2 =... = p n 1 := (0, 2) and p n := (2, 0). Let h(a, B) := sup a A (inf b B d(a, b)) for two non-empty subsets A, B of R 2, where d(a, b) denotes the standard Euclidean distance of a, b R 2. We denote the control polygon formed by p 0, p 1,..., p n by P n. Derive a formula (in dependence of n) for h(p n, B n ). Assignment 23 Consider a quadratic polynomial p: R R. Prove directly by establishing the three properties stated in Thm. 94, and without resorting to Lem. 95 or the like that f : R 2 R with ( ) x1 + x 2 f(x 1, x 2 ) := 2p p(x 1) 1 2 p(x 2) forms a polar form of p. Assignment 24 Consider a quadratic polynomial p: R R. Prove explicitly without simply citing Thm. 94, or resorting to Lem. 95 or the like that there exists a unique polar form for p. (You may use Ass. 23, even if you did not solve it.)

7 Homework Assignment Sheet VII (Due 22-Dec-2017) Assignment 25 We consider the bi-infinite knot vector τ := (t i ) i Z with t i := i for all i Z. Compute N 3,2 (t) and use this result to compute N 3,2(t) and N 3,2(t). What do we get as curvature of the graph (t, N 3,2 (t)) for t := 9 and t := 3? 2 Assignment 26 Consider a bi-infinite knot vector. Verify explicitly (without resorting to Lem. 121) that N i,k(t) = k k N i,k 1 (t) N i+1,k 1 (t) t i+k t i t i+k+1 t i+1 for k := 2, all i Z and all t ]t i, t i+1 [. Assignment 27 Let c R be arbitrary but fixed, and consider two bi-infinite knot vectors τ := (..., t 2, t 1, t 0, t 1, t 2,...) and τ := (..., t 2, t 1, t 0, t 1, t 2,...) with t j = t j + c for all j Z. Prove that N i,k,τ (t) = N i,k,τ (t + c) for all i Z and all k N 0. Assignment 28 Consider the (infinite) knot vector τ := (t 0, t 1, t 2, t 3, t 4,...) with t 0 := 0, t 1 := 1, t 2 = t 3 := 2 and t j := j 1 for all j N \ {1, 2, 3}. Identify all (permissible) values for i N 0 such that N i,1,τ (t) is not continuous.

8 Homework Assignment Sheet VIII (Due 12-Jan-2018) Assignment 29 Is a B-spline curve necessarily convex if its control polygon is convex? (We call a closed curve convex if it is simple and if the region bounded by the curve is convex; an open curve is convex if the closed curve formed by the original open curve and the straight-line segment between its start and end point is convex.) Assignment 30 Consider a quadratic B-spline P with three control points p 0, p 1, p 2 and knot vector τ := (0, 0, 0, 1, 1, 1). Prove explicitly that P is a Bézier curve. Assignment 31 Consider the clamped degree-two B-spline curve P defined by the control points ( ) ( ) ( ) ( ) ( ) ( ) ,,,,, and the knot vector τ := (0, 0, 0, 1 /4, 1 /2, 3 /4, 1, 1, 1). Does P pass through the point q := ( ) 1/2 1/2? Could we get different results for other clamped (possibly non-uniform) knot vectors instead of τ? Assignment 32 Consider the knot vector τ := (t 0, t 1, t 2, t 3, t 4, t 5, t 6, t 7, t 8, t 9 ), with t 0 = t 1 = t 2 < t 3 < t 4 = t 5 < t 6 < t 7 = t 8 = t 9. Prove explicitly (without resorting to Lem. 139) that P(t 4 ) = p 3 for a B-spline curve P of degree two and p 0,..., p 6 as control points.

9 Homework Assignment Sheet IX (Due 19-Jan-2018) Assignment 33 Let P be a B-spline curve of degree k with control points p 0, p 1,..., p n and knot vector τ := (t 0, t 1,..., t n+k+1 ). Let i, j N 0 with i n and j k. Let a N 0 such that t [t a, t a+1 [ for some arbitrary but fixed t [t k, t n+1 [. Prove that p i,j (t), as defined by de Boor s Algorithm (Thm. 140), is irrelevant for obtaining P(t) unless j {0, 1,..., k} and i {a k + j, a k + j + 1,..., a}. Assignment 34 Consider a clamped cubic B-spline curve P with uniform knot vector τ over the parameter range [0, 1] and control points p 0, p 1,..., p 6. Express P( 2 ) in terms of 5 p 0, p 1,..., p 6. (No need to recursively substitute into and simplify the resulting expression for P( 2).) 5 Assignment 35 Consider a clamped cubic B-spline curve P with uniform knot vector τ over the parameter range [0, 1] and control points p 0, p 1,..., p 7. Insert the new knot t := 1 into τ. 2 That is, compute a new knot vector and new control points such that the shape of P remains unchanged despite of the knot insertion. Assignment 36 Let n, m N and k, k N 0 with k n and k m. Furthermore, let σ := (s 0, s 1,..., s n+k +1) and τ := (t 0, t 1,..., t m+k +1) be two knot vectors. Let S be the B- spline surface relative to σ and τ with control net (p i,j ) n,m i,j=0, and let y R with 0 < y < 1. We assume that p i,j has i as its x-coordinate and j as its y-coordinate, for 0 i n and n m 0 j m. What is the intersection of S with a plane Π that is parallel to the xz-plane and that contains the point (0, y, 0)?

10 Homework Assignment Sheet X (Due 26-Jan-2018) Assignment 37 Use homogeneous coordinates and the projective plane to study the intersection of two lines, and compare this to the standard homogeneous coordinates. (Remember, the equation of a line in inhomogeneous coordinates is given by a x + b y + c = 0, for some a, b, c R.) What do you get for the intersection in (a) homogeneous coordinates, (b) inhomogeneous coordinates if the lines are (1) parallel, (2) not parallel? What do you get for the equation of the line passing through two points p 1, p 2 (in homogeneous coordinates)? Assignment 38 For a function f i : R 2 R in inhomogeneous coordinates, we obtain a function f h : R 2 (R \ {0}) R in homogeneous coordinates as f h (x, y, w) := f i ( x, y ). Prove: w w (1) If f i (x, y) = 0 is a bivariate polynomial equation, then f h (x, y, w) = 0 is a trivariate polynomial equation such that all monomials have the same degree. (2) The equation f i (x, y) = 0 has at least one solution if and only if f h (x, y, w) = 0 has infinitely many solutions. Assignment 39 Prove explicitly (without resorting to the interpretation of a NURBS curve as a B-spline curve in one dimensions higher) that a NURBS curve of degree k relative to a clamped knot vector τ := (t 0, t 1,..., t n+k+1 ) and control points p 0, p 1,..., p n starts in p 0, no matter which positive weights w 0, w 1,..., w n are chosen. Assignment 40 Can the relocation of one control point of a NURBS curve be compensated by an adjustment of some weights? Show that there exists NURBS curves for which even the modification of all weights does not suffice to compensate the relocation of one control point.

11 Homework Assignment Sheet XI (Due 02-Feb-2018) Assignment 41 Does the Hausdorff distance H form a metric on the set of all non-empty sub-sets of a metric space X? Assignment 42 Let I := [0, 1] and β, γ : I R n be two Bézier curves. Which of the following three claims is true? (1) H(β(I), γ(i)) Fr(β, γ). (2) H(β(I), γ(i)) Fr(β, γ). (3) neither (1) nor (2) need be correct. Assignment 43 For some n N 0, let q 0, q 1,..., q n be n + 1 points in R 2 and denote their coordinates by (x 0, y 0 ), (x 1, y 1 ),..., (x n, y n ). We assume that x i x j for all 0 i < j n. Prove directly (without resorting to Lagrange interpolation or the like): There exists exactly one polynomial p(x) of the form p(x) := a 0 + a 1 (x x 0 ) + a 2 (x x 0 )(x x 1 ) a n (x x 0 )(x x 1 )... (x x n 1 ), with a 0, a 1,..., a n R, such that p(x i ) = y i for all 0 i n. Assignment 44 For some n N 0, let q 0, q 1,..., q n be n + 1 points in R 2 and denote their coordinates by (x 0, y 0 ), (x 1, y 1 ),..., (x n, y n ). We assume that x i x j for all 0 i < j n. In the lecture I claimed that there exists exactly one polynomial p of degree n such that p(x i ) = y i for all 0 i < j n. Lagrange interpolation (Corr. 168) or Ass. 43 show that such a polynomial does indeed always exist. Prove directly that this polynomial is unique if its degree is allowed to be at most n.