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Lecture 11 - Spline curves (slides)

# Reversing the de casteljau algorithm graphics lecture

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Unformatted text preview: phics Lecture 11: Slide 30! Four point Bezier Curves and Cubic Patches Four point Bezier Curves and Cubic Patches " •  We can show their equivalence:  ! We can show their equivalence: Four point Bezier curve = Cubic patch going through the ﬁrst and last knots (P0 and P3 ) It is possible to show their equivalence by •  It is possible to show their equivalence by ! I Expanding the iterative blending equation ! –  Expanding the iterative blending equation I Reversing the de Casteljau algorithm ! –  Reversing the de Casteljau algorithm Graphics Lecture 11: Slide 31! 31 / 38 Expanding the blending equation Expanding the blending equation Expanding of fourblending equation " the knots we can Expanding the blending equationexpand the Bernstein blending For the case Forfunction to get a knots we can expand the Bernstein blending For the case of four knots we can expand the Bernstein blending the case of four polynomial in µ: function to get a polynomial in µ::we can expand the Bernstein •  For to get a polynomial in µ function the case of four knots 3 X blending3 function to get a polynomial in µ: ! X P(µ) = 3 X W (3, i, µ)Pi P(µ) = W (3, i, µ)P P(µ) = iW (3, i, µ)Pii =0 ii=0 ==0(1 3µ)3 P0 + 3µ(1 2µ)2 P1 +23µ2 (1 µ)P2 + µ3 P3 = (1 µ) 3P0 + 3µ(1 µ) 2P1 + 3µ 2(1 µ)P2 + µ3 P3 = (1 µ) P0 + 3µ(1 µ) P1 + 3µ (1 µ)P2 + µ3 P3 •This can multiplied outoutout give equation of the the form:   Thisbe bebe multiplied giveto giveequation of form: can multiplied to to an an an equation of the This can be multiplied out to give an equation of the form: This can form: ! P(µ) = a3 µ3 + a2 µ2 + a1 µ + a0 P(µ) = a3 µ3 + a2 µ2 + a1 µ + a0 P(µ) = a3 µ3 + a2 µ2 + a1 µ + a0 where ! w where here a0 = P 0 where a0 = P 0 a0 a = = 0 3 P P 3P0 1 1 a 1 = 3 P 1 3P0 a1 a = = P3P 3P6P + 3P 31 0 2 2 1 0 a 2 = 3 P 2 6P1 + 3 P 0 a2 a = = PP 6PP++PP 32 3 1 2 3 3 0 1 P0 3 3 a3 = P3 3P2 + 3P1 P0 a3 = P3 3P2 + 3P1 P0 32 / 38 Expanding the blending equation Expanding the blending equation " Expanding the blending equation These equations are linear These equations are linear •  These equations are a linear ! P 0= 0 ! a0 = P 0 3 P a= 1 ! ! a1 = a2 a2 = a 1 3P0 3 P 1 3 P P0 6P + 3 P 3 = 2 1 0 3P2 6P1 P 3+ 0 P 3+2 P 3 1 P0 3 = P3 a3 = P3 3P2 + 3P1 P0 Note that P0 P a P3 are the endpoints •  Note that P0 and and re the endpoints! NoteRecallP0 and P33 are the endpoints that the matrix form used for a cubic spline •  Recall the matrix form used for a cubic spline patchpatch ! Recall the matrix form used for a cubic spline patch ! 0 10 10 1 a 1 0 0 10 P 0 10 0 0 10 B a0 a1 C B 0 1 01 00 0 0 C B P0 C P0 BC C = B CB C0 C B a1 a A @ 3 @C 2 B 0 12 0 3 C1 A @ C 3 A 0 C B P0 P B B0 =B @ a2 A @ 3 2 21 3 2 A 1 P3 A 0 [email protected] a3 P3 a3 2 1 2 1 P0 3 Graphics see Lecture 11: Slide 33! see 33 / 38 Expanding the blending equation Expanding the blending equation " Expanding the blending equation These equations are linear These equations are linear •  These equations are = P! a0 linear 0 ! a0 = PP a = 30 1 ! ! 3P0 3P0 + 3 P 6P 3 P1 3 P2 1 0 3P2 36P1+ 33P0 P P3 P2 + P1 3 0 a3...
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