Calc1131W9T3 - de Casteljau Bezier curves

Tuesday, 1 May 2018

8:30 PM

Machine generated alternative text: Linear interpolation is a way of parametrising the line between two points. If A = [1, 1] and B = [2, 2] , then the linear interpolation is: p(t) = (1 — + tB, 0 s t S 1 where t represents time. The point p(t) moves with constant velocity from A to B. This construction is shown in the GeoGebra app below. 2.5 1.5 0.5 0.5 t=O.7 = (1.7 .7) 1.5 2.5 Use the t slider to find the points: • p(o.2) • p(0.4) • p(o.7) [1.2, 1.2] [1.7,1.7] Note: the Maple syntax for the point [1,

 

Machine generated alternative text: For thousands of years; people have used mathematics to describe curves in the plane. So it may come as a surprise that in the early 1960's Pierre Bezier and Paul de Casteljau, two French car engineers working respectively for Renault and Citroen, independently discovered a new method for describing curves' Pierre Bezier (1910 - 1999) Paul de Casteijau (born 1930) Given the control points A = [1, 1], B = [2, 2] and C = [3, 1] , we construct the associated quadratic de Casteljau-Bezier curve by successive linear interpolations. The linear de Casteljau-Bezier curve from A to B is the linear interpolation from A to B: Similarly, the linear de Casteljau-Bezier curve from B to C is the linear interpolation from B to C Note the syntax for q(t) is of the form [1 + t, 1 — t]. Once we have two linear de Casteljau-Bezier curves; we can linearly interpolate p(t) and q(t) (our linear interpolations') to create a quadratic de Casteljau-Bezier curve from A to C r(t) = (1 — + tq(t) = (1 - t)2A + 2t(1 - + Pc.

 

Machine generated alternative text: This construction is displayed in the GeoGebra app below: p(t) B Find (to two decimal places) the points t=093 r(t) = (2.86, 1.13) • r(0.3) • r(0.36) — • r(0.93) — [1 72, 1.46] [286, 1<3]

 

Machine generated alternative text: It is always possible to go deeper For example, a cubic de Casteljau Bezier curve is constructed by linear interpolation of two quadratic curves with two common control points The plot below shows a cubic de Casteljau Bezier curve (the locus of S) with control points A, B, C, D constructed from two quadratic curves with control points A, B, C and B, C, D respectively. t=042 s Note: the Maple syntax for the point [1, 2] is [1, 2]

 

Machine generated alternative text: Points on the line segment AB also have an interpretation as the centre of mass of two appropriate masses a and b at A, B. For example if we put a mass of 3 at A, and a mass of 1 at B i then the pont of balance will be According to Archimedes' law of the lever, the centre of mass of the rod AB is the point C , where Let's normalise this equation, so that the sum of the masses equals 1 Now if we let t then this equation becomes This equation can be re-arranged as OC Given the points A = [1, 1], B = [2, 2] with respective masses a Note: the Maple syntax for the point [1, 2] is [1, 2] aAC = bCB CB (1 - t)AC = tCB 4, the centre of mass is at the point [11/7: 11/7] 0 0B 3 and b C

 

Machine generated alternative text: The notion of centre of mass is related to linear; quadratic and even cubic de Casteljau - Bezier curves. Let A = [1,3] and C = [2, —2] and consider the (weightless) triangle ABC Add variable masses to the verticies of this triangle as follows: • Place mass m A = (1 —t) at vertex A • Place mass = 2t(1 — t) at vertex B • Place mass mc = t at vertex C The centre of mass of this triangle is: Hence Note: the Maple syntax for [0, 1] is [O, 1] 3 2 -2 = (1 — + 2t(1 - 0B + pc. 2

 

Machine generated alternative text: De Casteljau Bezier curves are important in design, as they allow us to specify a complicated curve by breaking it into simple pieces, and specifying those pieces by just a few control points each For example, in the diagram to specify the profile of the hood, we could join two quadratic de Casteljau-Bezier curves7 so that they are differentiable at the point at which they join 4. 1973 Citroen DS Special Consider two quadratic de Casteljau-Bezier curves: p(t) has control points Al , Bl, Cl and q(t) has control points .42, B2, C2 • Setting Cl = 242 ensures that the combined curve is continuous • The direction ofthe tangent top at Cl is the direction ofthe line • The direction ofthe tangent toq at 142 is the direction ofthe line O at Cl SICI A2B2

 

Machine generated alternative text: Hence the piecewise curve p (2t) r(t) q(2t- 1) is differentiable at Cl when the points Al are collinear 1 1

 

 

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