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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | --Copyright The Numerical Algorithms Group Limited 1994.
-- Create Affine transformations (DH matrices) that transform
-- a given triangle into another given triangle
-- tri2tri(t1, t2) returns a DHMATRIX which transforms t1 to t2,
-- where t1 and t2 are the vertices of two triangles in 3-space.
tri2tri(t1: List Point DoubleFloat, t2: List Point DoubleFloat): DHMATRIX(DoubleFloat) ==
n1 := triangleNormal(t1)
n2 := triangleNormal(t2)
tet2tet(concat(t1, n1), concat(t2, n2))
-- tet2tet(t1, t2) returns a DHMATRIX which transforms t1 to t2,
-- where t1 and t2 are the vertices of two tetrahedrons in 3-space.
tet2tet(t1: List Point DoubleFloat, t2: List Point DoubleFloat): DHMATRIX(DoubleFloat) ==
m1 := makeColumnMatrix t1
m2 := makeColumnMatrix t2
m2 * inverse(m1)
-- put the vertices of a tetrahedron into matrix form
makeColumnMatrix(t) ==
m := new(4,4,0)$DHMATRIX(DoubleFloat)
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
-- return a vector normal to the given triangle, whose length
-- is the square root of the area of the triangle
triangleNormal(t) ==
a := triangleArea t
p1 := t.2 - t.1
p2 := t.3 - t.2
c := cross(p1, p2)
len := length(c)
len = 0 => error "degenerate triangle!"
c := (1/len)*c
t.1 + sqrt(a) * c
-- compute the are of a triangle using Heron's formula
triangleArea t ==
a := length(t.2 - t.1)
b := length(t.3 - t.2)
c := length(t.1 - t.3)
s := (a+b+c)/2
sqrt(s*(s-a)*(s-b)*(s-c))
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