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1 Dr. Scott Schaefer Least Squares Curves, Rational Representations, Splines and Continuity

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2/72 Degree Reduction Given a set of coefficients for a Bezier curve of degree n+1, find the best set of coefficients of a Bezier curve of degree n that approximate that curve

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3/72 Degree Reduction

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4/72 Degree Reduction

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5/72 Degree Reduction

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6/72 Degree Reduction

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7/72 Degree Reduction

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8/72 Degree Reduction Problem: end-points are not interpolated

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9/72 Least Squares Optimization

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10/72 Least Squares Optimization

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11/72 Least Squares Optimization

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12/72 Least Squares Optimization

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13/72 Least Squares Optimization

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14/72 Least Squares Optimization

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15/72 The PseudoInverse What happens when isn’t invertible?

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16/72 The PseudoInverse What happens when isn’t invertible?

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17/72 The PseudoInverse What happens when isn’t invertible?

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18/72 The PseudoInverse What happens when isn’t invertible?

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19/72 The PseudoInverse What happens when isn’t invertible?

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20/72 The PseudoInverse What happens when isn’t invertible?

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21/72 The PseudoInverse What happens when isn’t invertible?

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22/72 The PseudoInverse What happens when isn’t invertible?

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23/72 The PseudoInverse What happens when isn’t invertible?

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24/72 The PseudoInverse What happens when isn’t invertible?

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25/72 The PseudoInverse What happens when isn’t invertible?

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26/72 The PseudoInverse What happens when isn’t invertible?

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27/72 The PseudoInverse What happens when isn’t invertible?

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28/72 The PseudoInverse What happens when isn’t invertible?

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29/72 Constrained Least Squares Optimization

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30/72 Constrained Least Squares Optimization Constraint Space Error Function F(x) Solution

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31/72 Constrained Least Squares Optimization

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32/72 Constrained Least Squares Optimization

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33/72 Constrained Least Squares Optimization

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34/72 Constrained Least Squares Optimization

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35/72 Constrained Least Squares Optimization

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36/72 Least Squares Curves

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37/72 Least Squares Curves

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38/72 Least Squares Curves

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39/72 Least Squares Curves

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40/72 Degree Reduction Problem: end-points are not interpolated

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41/72 Degree Reduction

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42/72 Degree Reduction

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43/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

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44/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

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45/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

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46/72 Rational Curves Curves defined in a higher dimensional space that are “projected” down

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47/72 Why Rational Curves? Conics

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48/72 Why Rational Curves? Conics

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49/72 Why Rational Curves? Conics

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50/72 Why Rational Curves? Conics

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51/72 Derivatives of Rational Curves

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52/72 Derivatives of Rational Curves

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53/72 Derivatives of Rational Curves

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54/72 Derivatives of Rational Curves

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55/72 Splines and Continuity C k continuity:

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56/72 Splines and Continuity C k continuity:

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57/72 Splines and Continuity C k continuity:

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58/72 Splines and Continuity C k continuity:

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59/72 Splines and Continuity C k continuity:

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60/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m

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61/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m C 0 : p n =q 0

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62/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m C 0 : p n =q 0 C 1 : n(p n -p n-1 )=m(q 1 -q 0 )

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63/72 Splines and Continuity Assume two Bezier curves with control points p 0,…,p n and q 0,…,q m C 0 : p n =q 0 C 1 : n(p n -p n-1 )=m(q 1 -q 0 ) C 2 : n(n-1)(p n -2p n-1 +p n-2 )=m(m-1)(q 0 -2q 1 +q 2 ) …

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64/72 Splines and Continuity Geometric Continuity A curve is G k if there exists a reparametrization such that the curve is C k

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65/72 Splines and Continuity Geometric Continuity A curve is G k if there exists a reparametrization such that the curve is C k

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66/72 Splines and Continuity Geometric Continuity A curve is G k if there exists a reparametrization such that the curve is C k

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67/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

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68/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

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69/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain C k continuity!!!

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70/72 Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain C k continuity!!! Difficult to keep track of all the constraints.

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71/72 B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control

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72/72 B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control

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