$SPAD/src/input scherk.input

Timothy Daly December 3, 2016

Abstract

Contents

2

1 License

— license —

--Copyright The Numerical Algorithms Group Limited 1991.

———-

— * —

)set break resume
)spool scherk.output
)set message test on
)set message auto off
)clear all

———-

Scherk's minimal surface.
Defined by:
   exp(z) * cos(x) = cos(y)
See: A comprehensive Introduction to Differential Geometry, Vol. 3,
by Michael Spivak, Publish Or Persih, Berkeley, 1979, pp 249-252.

Off set for a single piece of Scherk’s minimal surface — * —

--S 1 of 7
(xOffset, yOffset):DoubleFloat
--R
--R                                                                   Type: Void
--E 1

———-

DrawScherk’s minimal surface on an m by n patch.

— * —

--S 2 of 7
drawScherk(m,n) ==
  free xOffset, yOffset
  space := create3Space()$ThreeSpace(DoubleFloat)
  for i in 0..m-1 repeat
    xOffset := i*%pi
    for j in 0 .. n-1 repeat
      rem(i+j, 2) = 0 => 'iter
      yOffset := j*%pi
      drawOneScherk(space)
  makeViewport3D(space, "Scherk's Minimal Surface")
--R

--R                                                                   Type: Void
--E 2

———-

The four patches which make up a single piece of Scherk’s minimal surface.

— * —

--S 3 of 7
scherk1(u,v) ==
  x := cos(u)/exp(v)
  point [xOffset + acos(x), yOffset + u, v, abs(v)]
--R
--R                                                                   Type: Void
--E 3
--S 4 of 7
scherk2(u,v) ==
  x := cos(u)/exp(v)
  point [xOffset - acos(x), yOffset + u, v, abs(v)]
--R
--R                                                                   Type: Void
--E 4
--S 5 of 7
scherk3(u,v) ==
  x := exp(v) * cos(u)
  point [xOffset + u, yOffset + acos(x), v, abs(v)]
--R
--R                                                                   Type: Void
--E 5
--S 6 of 7
scherk4(u,v) ==
  x := exp(v) * cos(u)
  point [xOffset + u, yOffset - acos(x), v, abs(v)]
--R
--R                                                                   Type: Void
--E 6

———-

We draw the surface by breaking it into 4 patches, and drawing them into a single space. — * —

--S 7 of 7
drawOneScherk(s) ==
  makeObject(scherk1, -%pi/2..%pi/2, 0..%pi/2,  space == s, _
             var1Steps == 28, var2Steps == 28)
  makeObject(scherk2, -%pi/2..%pi/2, 0..%pi/2,  space == s, _
             var1Steps == 28, var2Steps == 28)

  makeObject(scherk3, -%pi/2..%pi/2, -%pi/2..0, space == s, _
             var1Steps == 28, var2Steps == 28)
  makeObject(scherk4, -%pi/2..%pi/2, -%pi/2..0, space == s, _
             var1Steps == 28, var2Steps == 28)
  void()
--R
--R                                                                   Type: Void
--E 7
)spool
)lisp (bye)

———-

References^{[1] nothing}

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