# 21.13 antoine.input¶

Draw Antoine’s Necklace. Antoine’s Necklace Thank you to Matthew Grayson at IBM’s T.J Watson Research Center for the idea.

)set expose add con DenavitHartenbergMatrix                Bring DH matrices into


the environment

torusRot: DHMATRIX(DFLOAT)                                 The  transformation for

drawing a sub ring

drawRings(n) ==                                            Draw Antoine’s Necklace with n

s := createThreeSpace()                                  levels of recursive subdivision

dh:DHMATRIX(DFLOAT) := identity()                        The number of subrings is 10n

drawRingsInner(s, n, dh)                                 Do the real work
makeViewport3D(s, “Antoine’s Necklace”)

In order to draw Antoine rings, we take one ring, scale it down to a smaller size, rotate it around its central axis, translate it to the edge of the larger ring and rotate it around the edge to a point corresponding to its count (there are 10 positions around the edge of the larger ring). For each of these new rings we recursively perform the operations, each ring becoming 10 smaller rings. Notice how the DHMATRIX operations are used to build up the proper matrix composing all these transformations.

drawRingsInner(s, n, dh) ==                                Recursively draw Antoine’s
n = 0 =>                                                 Necklace   drawRing(s, dh)   void()

t := 0.0@DFLOAT                                          Angle around ring

p := 0.0@DFLOAT                                          Angle of subring from plane

tr := 1.0@DFLOAT                                         Amount to translate subring

inc := 0.1@DFLOAT                                        The translation increment

for i in 1..10 repeat                                    Subdivide into 10 linked rings
tr := tr + inc inc := -inc dh’ := dh*rotatez(t)*translate(tr,0.0@DFLOAT,0.0@DFLOAT)*
Transform ring in center
to a link          rotatey(p)*scale(0.35@DFLOAT0.48@DFLOAT0.4@DFLOAT)   drawRingsInner(s, n-1, dh’)   t := t + 36.0@DFLOAT   p := p + 90.0@DFLOAT void()

drawRing(s, dh) ==                                         Draw a single ring into

free torusRot                                            the given subspace,

torusRot := dh                                           transformed by the given
DHMATRIX makeObject(torus, 0..2*%pi, 0..2*%pi, var1Steps == 6,            space == s, var2Steps == 15)

torus(u ,v) ==                                             Parameterization of a torus,

cu := cos(u)/6                                           transformed by the

DHMATRIX in torusRot.
torusRot*point [(1+cu)*cos(v),(1+cu)*sin(v),(sin u)/6]