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This is the Past Exam Paper of Math Tripos which includes Geometry of 3-Dimensional Manifolds, Terms Nilpotent Group, Normal Abelian Subgroup, Nilpotent Lie Group, Spectral Sequence, Group Extension, Proof of Loop Theorem, Surface Fibration etc. Key important points are: Computer-Aided Geometric Design, Euclidean Space, Particular Configurations, Bernstein Polynomials, Bezier Curve, Parametric Curve, Convex Hull of Control Points, General Subdivision Curve, Univariate Box-Splines
Typology: Exams
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Tuesday 3 June 2003 9 to 11
Attempt FOUR questions. There are six questions in total. The questions carry equal weight.
1 AB and CD are line segments in Euclidean space of 3 dimensions. Define an algorithm for determining points P (on AB) and Q (on CD) such that the distance P Q is a minimum. You may assume that the endpoints A, B, C and D of the two segments are in general position. What particular configurations would need to be considered, if the endpoints could not be assumed to be in general position?
2 What are the Bernstein polynomials of degree n? A Bezier curve is a parametric curve whose equation is
P (t) = ∑^ n i=
Pifi(t)
where the Pi are the control points and the fi are Bernstein polynomials. Show that the piece of this curve lying between t = 0 and t = 1 lies inside the convex hull of the control points, and that the first derivative of the curve can be expressed as a similar combination of the first differences of the control points and the Bernstein polynomials of one degree lower.
3 What are the enquiries which can be made of a general parametric curve? Define one algorithm to compute, in terms of those enquiries, the intersection of a general subdivision curve with a given plane. What are the strengths and weaknesses of your chosen algorithm?
4 What are the enquiries which can be made of a general subdivision curve? Define an algorithm to compute, in terms of those enquiries, the intersections of a general parametric curve with a given plane. How is good speed obtained in such an algorithm?
Paper 68
