CS 410 Hermite Curve Example LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElUm9zc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZPLUYsNiZRKkJldmVyaWRnZUYnRi9GMkY1Rjw= November 2007 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Below are initializations, typically hidden. with(LinearAlgebra): readlib(unassign): unassign('Q','x1','y1','z1','x2','y2','z2','dx1','dy1','dz1','dx2','dy2','dz2'): unassign('G','M','T'):

Here are the basis, geometry matrices and curve equations for the Hermite Cuve. Note this form is essentially the transpose of the development in the PowerPoint notes. MM := Matrix(4, 4, [[2, -3, 0, 1], [-2, 3, 0, 0], [1, -2, 1, 0], [1, -1, 0, 0]]): PP1 := Matrix(3, 1, [x1, y1, z1]): PP2 := Matrix(3, 1, [x2, y2, z2]): RR1 := Matrix(3, 1, [dx1, dy1, dz1]): RR2 := Matrix(3, 1, [dx2, dy2, dz2]): GG := Matrix(3, 4, [[x1, x2, dx1, dx2], [y1, y2, dy1, dy2], [z1, z2, dz1, dz2]]): TT := Matrix(4, 1, [t^3, t^2, t, 1]): Qt1 := evalm(GG &* MM &* TT): Qt1[1,1] := sort(collect(expand(Qt1[1,1]),{x1,x2,dx1,dx2}),[x1,x2,dx1,dx2]): Qt1[2,1] := sort(collect(expand(Qt1[2,1]),{y1,y2,dy1,dy2}),[y1,y2,dy1,dy2]): Qt1[3,1] := sort(collect(expand(Qt1[3,1]),{z1,z2,dz1,dz2}),[z1,z2,dz1,dz2]): # This mapping expedites the plotting below xp := t -> eval(Qt1[1,1]): yp := t -> eval(Qt1[2,1]): zp := t -> eval(Qt1[3,1]): # Here is the output Q(t) = G * M * T; Q(t) = evalm(GG) * evalm(MM) * evalm(TT); Q(t) = evalm(Qt1);

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For the sake of completeness, here is the tranposed equivalent that matches what is in the class PowerPoint notes. Q(t) = T^T*M^T*G^T; Q(t) = evalm(Transpose(TT)) * evalm(Transpose(MM)) * evalm(Transpose(GG)); unassign('Qt2'): Qt2 := evalm(Transpose(TT) &* Transpose(MM) &* Transpose(GG)): Qt2[1, 1] := sort(collect(expand(Qt2[1, 1]), {x2, dx1, dx2, x1}), [x1, x2, dx1, dx2]): Qt2[1, 2] := sort(collect(expand(Qt2[1, 2]), {dy2, dy1, y1, y2}), [y1, y2, dy1, dy2]): Qt2[1, 3] := sort(collect(expand(Qt2[1, 3]), {dz1, dz2, z1, z2}), [z1, z2, dz1, dy2]): Qt2 := convert(Qt2, Matrix): Q(t)^T = Transpose(Qt2);

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JSFH

Consider one of the simplest cases, a Hermite Curve can be directed to follow a straight path. 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x1 := GT[1, 1]: y1 := GT[1, 2]: z1 := GT[1, 3]: x2 := GT[2, 1]: y2 := GT[2, 2]: z2 := GT[2, 3]: dx1 := GT[3, 1]: dy1 := GT[3, 2]: dz1 := GT[3, 3]: dx2 := GT[4, 1]: dy2 := GT[4, 2]: dz2 := GT[4, 3]: plot([xp(t), yp(t), t=0..1]);

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Consider one of the simplest cases, a Hermite Curve can be directed to follow a straight path. 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

x1 := GT[1, 1]: y1 := GT[1, 2]: z1 := GT[1, 3]: x2 := GT[2, 1]: y2 := GT[2, 2]: z2 := GT[2, 3]: dx1 := GT[3, 1]: dy1 := GT[3, 2]: dz1 := GT[3, 3]: dx2 := GT[4, 1]: dy2 := GT[4, 2]: dz2 := GT[4, 3]: plot([xp(t), yp(t), t=0..1]);

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

JSFH

Consider one of the simplest cases, a Hermite Curve can be directed to follow a straight path. 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

x1 := GT[1, 1]: y1 := GT[1, 2]: z1 := GT[1, 3]: x2 := GT[2, 1]: y2 := GT[2, 2]: z2 := GT[2, 3]: dx1 := GT[3, 1]: dy1 := GT[3, 2]: dz1 := GT[3, 3]: dx2 := GT[4, 1]: dy2 := GT[4, 2]: dz2 := GT[4, 3]: plot([xp(t), yp(t), t=0..1]);

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

JSFH

Consider one of the simplest cases, a Hermite Curve can be directed to follow a straight path. 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

x1 := GT[1, 1]: y1 := GT[1, 2]: z1 := GT[1, 3]: x2 := GT[2, 1]: y2 := GT[2, 2]: z2 := GT[2, 3]: dx1 := GT[3, 1]: dy1 := GT[3, 2]: dz1 := GT[3, 3]: dx2 := GT[4, 1]: dy2 := GT[4, 2]: dz2 := GT[4, 3]: plot([xp(t), yp(t), t=0..1]);

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JSFH

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