Lecture 11 - Spline curves (slides)

A3 3 aa2 2 aa1 a0 a each patch the form each patch

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Unformatted text preview: ol points Calculating a CubicCubic Spline Patch Calculating a Spline Patch Calculating Calculating a Cubic Spline Patch a Cubic Spline Patch" Each patch has thehas the P = !a3 µ3= aa2 µ2+ aa1 µ + a0 µ + a Each patch the form: •  Each patch hasform:form: P + µ3 + µ2 + a 3 2 1 3 0 Each patch has the form: P = a3 µ + a2 µ2 + a1 µ + a0 For theFor the which joins points points P iand we have have patch patch which joins Pi and P +1 , P , we i i • I the patch which joins points Pi and Pi+1+1 have ! For , we µ = I at = i at P the patch which joins points Pi and Pi+1 , we have 0 µ P 0 For i I µ = I at Pi+1 1 µ = 1 at P µ = 0 at Pi I +1 i µ = 1 at Pi+1 Substituting these values values we get Substituting these we get •  Substituting these values we get ! Substituting these values we get I Pi = Pi 0 = a0 a Pi+1 Pi+1 3 = a2 3 + 1i2 + 0 1 a0 a0 = a + a+ a a+ =a + Pa Pi+1 = a3 + a2 + a1 + a0 Graphics Lecture 11: Slide 17! 17 / 38 17 / Calculating a Cubic Spline Patch Calculating a Cubic Spline Patch Calculating a Cubic Spline Patch Calculating a Cubic Spline Patch" Di ↵erentiating P = P3=3a µ32 +2a µ21+ + a0+ a get eget ! Di↵erentiating a µ + a µ + a µ a a0 we get •↵Differentiating a3 µ!3 + a2 µ2 + a1 µ + 1 µ we 0 we get !2! ! w 3 Di erentiating P = P00 =P0 a3 µ2a+µ2a2 µ a a1+ a 3 = 32 2 + 2 + µ 3 1 P = 3 a3 µ + 2 a2 µ +2a1 Substituting µ = 0 at 0 i and µ = 1 at 1 i+1Pwe , we get† Pat P and µ = Pat , get†† Substituting µ µ i+1 Substituting µ = for= Pi 0 at iPi =ndat = i1 a,t we get e get! •  Substituting 0 at = and µ a 1 µ P +1 Pi+1 w P00 P0 i= i P 0i = P0i+1P0= i+1 P = i+1 † Graphics Lecture 11: Slide 18! Remember that P’s † † a1 a a= 1 1 3 a3 +a a2 +a 1+ a = + 23a++ a2 32 2a 1 3a 3 2 and a’s are all vectors Remember that P’s and a’s are all vectors Remember that P’s and a’s are all vectors 1 18 / 38 18 / 38 18 / 38 Calculating a Cubic Spline Patch Calculating a Cubic Spline Patch" •  Putting these equations into matrix form we get: Putting these four four equations into matrix form we get:! 0 1 B0 B @1 0 0 1 1 1 0 0 1 2 0 0 1 3 10 1 0 1 a0 Pi C Ba1 C B P0 C CB C = B i C A @a2 A @Pi+1 A a3 P0 +1 i TheThe initial matrix is always the whether the points Ppoints P •  initial matrix is always the same same whether the are in 2-Daor in 3-D or in 3-D ! re in 2-D ! Graphics Lecture 11: Slide 19! 19 / 38 Calculating a Cubic Spline Patch" Calculating a Cubic Spline Patch •  Finally, inverting the matrix gives us the values of a0, . . . , Finally, inverting the matrix gives us the values of a0 , . . . , a3 that a3 that we want ! we want 0 1 0 a0 B a1 C B B CB @ a2 A = @ a3 1 0 3 2 0 1 2 1 0 0 3 2 •  Notice that matrix is is same Notice that thethe matrixthe the same ! –  for every patch! I for every patch –  whether the data are 2-D, 3-D, ... ! I whether the data are 2-D, 3-D, ... Graphics Lecture 11: Slide 20! . 10 1 0 Pi 0 C B P0 C iC CB 1 A @ Pi+1 A 1 P0 +1 i B...
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This document was uploaded on 03/26/2014.

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