Section21.11

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## 21.11 dhtri.input

Create affine transformations (DH matrices) that transform a given triangle into another.

tri2tri: (List Point DFLOAT, List Point DFLOAT) -> DHMATRIX(DFLOAT)
Compute a DHMATRIX that
tri2tri(t1, t2) == transforms t1 to t2, where
n1 := triangleNormal(t1) t1 and t2 are the vertices
n2 := triangleNormal(t2) of two triangles in 3-space
tet2tet(concat(t1, n1), concat(t2, n2))
tet2tet: (List Point DFLOAT, List Point DFLOAT) -> DHMATRIX(DFLOAT)
Compute a DHMATRIX that
tet2tet(t1, t2) == transforms t1 to t2,
m1 := makeColumnMatrix t1 where t1 and t2 are the
m2 := makeColumnMatrix t2 vertices of two tetrahedrons
m2 * inverse(m1) in 3-space
makeColumnMatrix(t) == Put the vertices of a tetra-
m := new(4,4,0)$DHMATRIX(DFLOAT) hedron into matrix form
for x in t for i in 1..repeat
for j in 1..3 repeat
m(j,i) := x.j
m(4,i) := 1
m
triangleNormal(t) == Compute a vector normal to
a := triangleArea t the given triangle, whose
p1 := t.2 - t.1 length is the square root
p2 := t.3 - t.2 of the area of the triangle
c := cross(p1, p2)
len := length(c)
len = 0 => error "degenerate triangle!"
c := (1/len)*c
t.1 + sqrt(a) * c
triangleArea t == Compute the area of a
a := length(t.2 - t.1) triangle using Heron's
b := length(t.3 - t.2) formula
c := length(t.1 - t.3)
s := (a+b+c)/2
sqrt(s*(s-a)*(s-b)*(s-c))
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