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(* $Id: morph3d.ml,v 1.1 2003/09/25 13:54:10 raffalli Exp $ *)
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* morph3d.c - Shows 3D morphing objects (TK Version)
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* This program was inspired on a WindowsNT(R)'s screen saver. It was written
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* from scratch and it was not based on any other source code.
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* Porting it to xlock (the final objective of this code since the moment I
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* decided to create it) was possible by comparing the original Mesa's gear
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* demo with it's ported version, so thanks for Danny Sung for his indirect
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* help (look at gear.c in xlock source tree). NOTE: At the moment this code
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* was sent to Brian Paul for package inclusion, the XLock Version was not
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* available. In fact, I'll wait it to appear on the next Mesa release (If you
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* are reading this, it means THIS release) to send it for xlock package
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* inclusion). It will probably there be a GLUT version too.
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* Thanks goes also to Brian Paul for making it possible and inexpensive
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* to use OpenGL at home.
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* Since I'm not a native english speaker, my apologies for any gramatical
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* My e-mail addresses are
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* marcelo@venus.rdc.puc-rio.br
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* Marcelo F. Vianna (Feb-13-1997)
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This document is VERY incomplete, but tries to describe the mathematics used
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in the program. At this moment it just describes how the polyhedra are
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generated. On futhurer versions, this document will be probabbly improved.
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Since I'm not a native english speaker, my apologies for any gramatical
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Marcelo Fernandes Vianna
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- Undergraduate in Computer Engeneering at Catholic Pontifical University
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- of Rio de Janeiro (PUC-Rio) Brasil.
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- e-mail: vianna@cat.cbpf.br or marcelo@venus.rdc.puc-rio.br
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For the purpose of this program it's not sufficient to know the polyhedra
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vertexes coordinates. Since the morphing algorithm applies a nonlinear
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transformation over the surfaces (faces) of the polyhedron, each face has
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to be divided into smaller ones. The morphing algorithm needs to transform
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each vertex of these smaller faces individually. It's a very time consoming
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In order to reduce calculation overload, and since all the macro faces of
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the polyhedron are transformed by the same way, the generation is made by
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creating only one face of the polyhedron, morphing it and then rotating it
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around the polyhedron center.
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What we need to know is the face radius of the polyhedron (the radius of
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the inscribed sphere) and the angle between the center of two adjacent
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faces using the center of the sphere as the angle's vertex.
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The face radius of the regular polyhedra are known values which I decided
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to not waste my time calculating. Following is a table of face radius for
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the regular polyhedra with edge length = 1:
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TETRAHEDRON : 1/(2*sqrt(2))/sqrt(3)
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OCTAHEDRON : 1/sqrt(6)
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DODECAHEDRON : T^2 * sqrt((T+2)/5) / 2 -> where T=(sqrt(5)+1)/2
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ICOSAHEDRON : (3*sqrt(3)+sqrt(15))/12
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I've not found any reference about the mentioned angles, so I needed to
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calculate them, not a trivial task until I figured out how :)
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Curiously these angles are the same for the tetrahedron and octahedron.
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A way to obtain this value is inscribing the tetrahedron inside the cube
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by matching their vertexes. So you'll notice that the remaining unmatched
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vertexes are in the same straight line starting in the cube/tetrahedron
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center and crossing the center of each tetrahedron's face. At this point
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it's easy to obtain the bigger angle of the isosceles triangle formed by
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the center of the cube and two opposite vertexes on the same cube face.
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The edges of this triangle have the following lenghts: sqrt(2) for the base
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and sqrt(3)/2 for the other two other edges. So the angle we want is:
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+-----------------------------------------------------------+
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| 2*ARCSIN(sqrt(2)/sqrt(3)) = 109.47122063449069174 degrees |
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+-----------------------------------------------------------+
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For the cube this angle is obvious, but just for formality it can be
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easily obtained because we also know it's isosceles edge lenghts:
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sqrt(2)/2 for the base and 1/2 for the other two edges. So the angle we
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+-----------------------------------------------------------+
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| 2*ARCSIN((sqrt(2)/2)/1) = 90.000000000000000000 degrees |
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+-----------------------------------------------------------+
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For the octahedron we use the same idea used for the tetrahedron, but now
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we inscribe the cube inside the octahedron so that all cubes's vertexes
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matches excatly the center of each octahedron's face. It's now clear that
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this angle is the same of the thetrahedron one:
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+-----------------------------------------------------------+
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| 2*ARCSIN(sqrt(2)/sqrt(3)) = 109.47122063449069174 degrees |
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+-----------------------------------------------------------+
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For the dodecahedron it's a little bit harder because it's only relationship
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with the cube is useless to us. So we need to solve the problem by another
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way. The concept of Face radius also exists on 2D polygons with the name
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Edge Radius For Pentagon (ERp)
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ERp = (1/2)/TAN(36 degrees) * VRp = 0.6881909602355867905
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(VRp is the pentagon's vertex radio).
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Face Radius For Dodecahedron
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FRd = T^2 * sqrt((T+2)/5) / 2 = 1.1135163644116068404
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Why we need ERp? Well, ERp and FRd segments forms a 90 degrees angle,
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completing this triangle, the lesser angle is a half of the angle we are
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looking for, so this angle is:
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+-----------------------------------------------------------+
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| 2*ARCTAN(ERp/FRd) = 63.434948822922009981 degrees |
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+-----------------------------------------------------------+
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For the icosahedron we can use the same method used for dodecahedron (well
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the method used for dodecahedron may be used for all regular polyhedra)
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Edge Radius For Triangle (this one is well known: 1/3 of the triangle height)
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ERt = sin(60)/3 = sqrt(3)/6 = 0.2886751345948128655
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Face Radius For Icosahedron
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FRi= (3*sqrt(3)+sqrt(15))/12 = 0.7557613140761707538
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+-----------------------------------------------------------+
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| 2*ARCTAN(ERt/FRi) = 41.810314895778596167 degrees |
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+-----------------------------------------------------------+
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let vect_mul (x1,y1,z1) (x2,y2,z2) =
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(y1 *. z2 -. z1 *. y2, z1 *. x2 -. x1 *. z2, x1 *. y2 -. y1 *. x2)
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(* Increasing this values produces better image quality, the price is speed. *)
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(* Very low values produces erroneous/incorrect plotting *)
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let tetradivisions = 23
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let cubedivisions = 20
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let octadivisions = 21
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let dodecadivisions = 10
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let icodivisions = 15
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let tetraangle = 109.47122063449069174
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let cubeangle = 90.000000000000000000
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let octaangle = 109.47122063449069174
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let dodecaangle = 63.434948822922009981
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let icoangle = 41.810314895778596167
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let sqrt15 = sqrt 15.
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let cossec36_2 = 0.8506508083520399322
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let cosd x = cos (float x /. 180. *. pi)
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let sind x = sin (float x /. 180. *. pi)
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(*************************************************************************)
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let front_shininess = 60.0
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let front_specular = 0.7, 0.7, 0.7, 1.0
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let ambient = 0.0, 0.0, 0.0, 1.0
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let diffuse = 1.0, 1.0, 1.0, 1.0
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let position0 = 1.0, 1.0, 1.0, 0.0
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let position1 = -1.0,-1.0, 1.0, 0.0
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let lmodel_ambient = 0.5, 0.5, 0.5, 1.0
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let lmodel_twoside = true
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let materialRed = 0.7, 0.0, 0.0, 1.0
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let materialGreen = 0.1, 0.5, 0.2, 1.0
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let materialBlue = 0.0, 0.0, 0.7, 1.0
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let materialCyan = 0.2, 0.5, 0.7, 1.0
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let materialYellow = 0.7, 0.7, 0.0, 1.0
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let materialMagenta = 0.6, 0.2, 0.5, 1.0
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let materialWhite = 0.7, 0.7, 0.7, 1.0
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let materialGray = 0.2, 0.2, 0.2, 1.0
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let all_gray = Array.create 20 materialGray
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let vertex ~xf ~yf ~zf ~ampvr2 =
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let xa = xf +. 0.01 and yb = yf +. 0.01 in
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let xf2 = sqr xf and yf2 = sqr yf in
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let factor = 1. -. (xf2 +. yf2) *. ampvr2
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and factor1 = 1. -. (sqr xa +. yf2) *. ampvr2
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and factor2 = 1. -. (xf2 +. sqr yb) *. ampvr2 in
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let vertx = factor *. xf and verty = factor *. yf
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and vertz = factor *. zf in
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let neiax = factor1 *. xa -. vertx and neiay = factor1 *. yf -. verty
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and neiaz = factor1 *. zf -. vertz and neibx = factor2 *. xf -. vertx
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and neiby = factor2 *. yb -. verty and neibz = factor2 *. zf -. vertz in
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GlDraw.normal3 (vect_mul (neiax, neiay, neiaz) (neibx, neiby, neibz));
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GlDraw.vertex3 (vertx, verty, vertz)
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let triangle ~edge ~amp ~divisions ~z =
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let divi = float divisions in
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let vr = edge *. sqrt3 /. 3. in
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let ampvr2 = amp /. sqr vr
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and zf = edge *. z in
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let ax = edge *. (0.5 /. divi)
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and ay = edge *. (-0.5 *. sqrt3 /. divi)
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and bx = edge *. (-0.5 /. divi) in
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for ri = 1 to divisions do
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GlDraw.begins `triangle_strip;
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for ti = 0 to ri - 1 do
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~xf:(float (ri-ti) *. ax +. float ti *. bx)
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~yf:(vr +. float (ri-ti) *. ay +. float ti *. ay);
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~xf:(float (ri-ti-1) *. ax +. float ti *. bx)
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~yf:(vr +. float (ri-ti-1) *. ay +. float ti *. ay)
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vertex ~xf:(float ri *. bx) ~yf:(vr +. float ri *. ay) ~zf ~ampvr2;
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let square ~edge ~amp ~divisions ~z =
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let divi = float divisions in
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and ampvr2 = amp /. sqr (edge *. sqrt2 /. 2.) in
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for yi = 0 to divisions - 1 do
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let yf = edge *. (-0.5 +. float yi /. divi) in
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let y = yf +. 1.0 /. divi *. edge in
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GlDraw.begins `quad_strip;
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for xi = 0 to divisions do
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let xf = edge *. (-0.5 +. float xi /. divi) in
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vertex ~xf ~yf:y ~zf ~ampvr2;
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vertex ~xf ~yf ~zf ~ampvr2
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let pentagon ~edge ~amp ~divisions ~z =
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let divi = float divisions in
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and ampvr2 = amp /. sqr(edge *. cossec36_2) in
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~f:(fun fi -> -. cos (float fi *. 2. *. pi /. 5. +. pi /. 10.)
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/. divi *. cossec36_2 *. edge)
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~f:(fun fi -> sin (float fi *. 2. *. pi /. 5. +. pi /. 10.)
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/. divi *. cossec36_2 *. edge)
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for ri = 1 to divisions do
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GlDraw.begins `triangle_strip;
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for ti = 0 to ri-1 do
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~xf:(float(ri-ti) *. x.(fi) +. float ti *. x.(fi+1))
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~yf:(float(ri-ti) *. y.(fi) +. float ti *. y.(fi+1));
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~xf:(float(ri-ti-1) *. x.(fi) +. float ti *. x.(fi+1))
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~yf:(float(ri-ti-1) *. y.(fi) +. float ti *. y.(fi+1))
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vertex ~xf:(float ri *. x.(fi+1)) ~yf:(float ri *. y.(fi+1)) ~zf ~ampvr2;
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let call_list list color =
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GlLight.material ~face:`both (`diffuse color);
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let draw_tetra ~amp ~divisions ~color =
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let list = GlList.create `compile in
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triangle ~edge:2.0 ~amp ~divisions ~z:(0.5 /. sqrt6);
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call_list list color.(0);
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GlMat.rotate ~angle:180.0 ~z:1.0 ();
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GlMat.rotate ~angle:(-.tetraangle) ~x:1.0 ();
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call_list list color.(1);
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GlMat.rotate ~angle:180.0 ~y:1.0 ();
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GlMat.rotate ~angle:(-180.0 +. tetraangle) ~x:0.5 ~y:(sqrt3 /. 2.) ();
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call_list list color.(2);
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GlMat.rotate ~angle:180.0 ~y:1.0 ();
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GlMat.rotate ~angle:(-180.0 +. tetraangle) ~x:0.5 ~y:(-.sqrt3 /. 2.) ();
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call_list list color.(3);
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let draw_cube ~amp ~divisions ~color =
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let list = GlList.create `compile in
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square ~edge:2.0 ~amp ~divisions ~z:0.5;
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call_list list color.(0);
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GlMat.rotate ~angle:cubeangle ~x:1.0 ();
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call_list list color.(i)
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GlMat.rotate ~angle:cubeangle ~y:1.0 ();
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call_list list color.(4);
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GlMat.rotate ~angle:(2.0 *. cubeangle) ~y:1.0 ();
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call_list list color.(5);
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let draw_octa ~amp ~divisions ~color =
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let list = GlList.create `compile in
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triangle ~edge:2.0 ~amp ~divisions ~z:(1.0 /. sqrt6);
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GlMat.rotate ~angle:180.0 ~y:1.0 ();
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GlMat.rotate ~angle:(-.octaangle) ~x:0.5 ~y ();
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call_list list color.(i);
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call_list list color.(0);
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GlMat.rotate ~angle:180.0 ~z:1.0 ();
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GlMat.rotate ~angle:(-180.0 +. octaangle) ~x:1.0 ();
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call_list list color.(1);
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List.iter [2, sqrt3 /. 2.0; 3, -.sqrt3 /. 2.0] ~f:do_list;
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GlMat.rotate ~angle:180.0 ~x:1.0 ();
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GlLight.material ~face:`both (`diffuse color.(4));
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GlMat.rotate ~angle:180.0 ~z:1.0 ();
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GlMat.rotate ~angle:(-180.0 +. octaangle) ~x:1.0 ();
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GlLight.material ~face:`both (`diffuse color.(5));
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List.iter [6, sqrt3 /. 2.0; 7, -.sqrt3 /. 2.0] ~f:do_list;
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let draw_dodeca ~amp ~divisions ~color =
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let tau = (sqrt5 +. 1.0) /. 2.0 in
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let list = GlList.create `compile in
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pentagon ~edge:2.0 ~amp ~divisions
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~z:(sqr(tau) *. sqrt ((tau+.2.0)/.5.0) /. 2.0);
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let do_list (i,angle,x,y) =
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GlMat.rotate ~angle:angle ~x ~y ();
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call_list list color.(i);
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call_list list color.(0);
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GlMat.rotate ~angle:180.0 ~z:1.0 ();
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[ 1, -.dodecaangle, 1.0, 0.0;
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2, -.dodecaangle, cos72, sin72;
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3, -.dodecaangle, cos72, -.sin72;
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4, dodecaangle, cos36, -.sin36;
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5, dodecaangle, cos36, sin36 ];
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GlMat.rotate ~angle:180.0 ~x:1.0 ();
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call_list list color.(6);
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GlMat.rotate ~angle:180.0 ~z:1.0 ();
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[ 7, -.dodecaangle, 1.0, 0.0;
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8, -.dodecaangle, cos72, sin72;
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9, -.dodecaangle, cos72, -.sin72;
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10, dodecaangle, cos36, -.sin36 ];
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GlMat.rotate ~angle:dodecaangle ~x:cos36 ~y:sin36 ();
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call_list list color.(11);
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let draw_ico ~amp ~divisions ~color =
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let list = GlList.create `compile in
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triangle ~edge:1.5 ~amp ~divisions
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~z:((3.0 *. sqrt3 +. sqrt15) /. 12.0);
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GlMat.rotate ~angle:180.0 ~y:1.0 ();
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GlMat.rotate ~angle:(-180.0 +. icoangle) ~x:0.5 ~y:(sqrt3/.2.0) ();
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call_list list color.(i)
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GlMat.rotate ~angle:180.0 ~y:1.0 ();
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GlMat.rotate ~angle:(-180.0 +. icoangle) ~x:0.5 ~y:(-.sqrt3/.2.0) ();
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call_list list color.(i)
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GlMat.rotate ~angle:180.0 ~z:1.0 ();
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GlMat.rotate ~angle:(-.icoangle) ~x:1.0 ();
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call_list list color.(i)
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call_list list color.(0);
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GlMat.rotate ~angle:180.0 ~x:1.0 ();
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call_list list color.(10);
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class view = object (self)
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val mutable smooth = true
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val mutable step = 0.
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val mutable draw_object = fun ~amp -> ()
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val mutable magnitude = 0.
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val mutable my_width = 640
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val mutable my_height = 480
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method width = my_width
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method height = my_height
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let ratio = float self#height /. float self#width in
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GlClear.clear [`color;`depth];
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GlMat.translate () ~z:(-10.0);
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GlMat.scale () ~x:(scale *. ratio) ~y:scale ~z:scale;
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~x:(2.5 *. ratio *. sin (step *. 1.11))
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~y:(2.5 *. cos (step *. 1.25 *. 1.11));
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GlMat.rotate ~angle:(step *. 100.) ~x:1.0 ();
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GlMat.rotate ~angle:(step *. 95.) ~y:1.0 ();
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GlMat.rotate ~angle:(step *. 90.) ~z:1.0 ();
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draw_object ~amp:((sin step +. 1.0/.3.0) *. (4.0/.5.0) *. magnitude);
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method reshape ~w ~h =
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GlDraw.viewport ~x:0 ~y:0 ~w:self#width ~h:self#height;
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GlMat.mode `projection;
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GlMat.load_identity();
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GlMat.frustum ~x:(-1.0, 1.0) ~y:(-1.0, 1.0) ~z:(5.0, 15.0);
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GlMat.mode `modelview
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method keyboard key =
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match (char_of_int key) with
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| _ -> match key with
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| 10(*return*) -> smooth <- not smooth
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| 27(*escape*) -> exit 0
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draw_object <- draw_tetra
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~divisions:tetradivisions
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~color:[|materialRed; materialGreen;
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materialBlue; materialWhite|];
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draw_object <- draw_cube
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~divisions:cubedivisions
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~color:[|materialRed; materialGreen; materialCyan;
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materialMagenta; materialYellow; materialBlue|];
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draw_object <- draw_octa
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~divisions:octadivisions
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~color:[|materialRed; materialGreen; materialBlue;
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materialWhite; materialCyan; materialMagenta;
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materialGray; materialYellow|];
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draw_object <- draw_dodeca
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~divisions:dodecadivisions
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~color:[|materialRed; materialGreen; materialCyan;
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materialBlue; materialMagenta; materialYellow;
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materialGreen; materialCyan; materialRed;
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materialMagenta; materialBlue; materialYellow|];
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draw_object <- draw_ico
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~divisions:icodivisions
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~color:[|materialRed; materialGreen; materialBlue;
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materialCyan; materialYellow; materialMagenta;
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materialRed; materialGreen; materialBlue;
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materialWhite; materialCyan; materialYellow;
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materialMagenta; materialRed; materialGreen;
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materialBlue; materialCyan; materialYellow;
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materialMagenta; materialGray|];
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GlDraw.shade_model (if smooth then `smooth else `flat)
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List.iter ~f:print_string
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[ "Morph 3D - Shows morphing platonic polyhedra\n";
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"Author: Marcelo Fernandes Vianna (vianna@cat.cbpf.br)\n";
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"Ported to LablGL by Jacques Garrigue\n";
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"Ported to lablglut by Issac Trotts\n\n";
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" [1] - Tetrahedron\n";
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" [2] - Hexahedron (Cube)\n";
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" [3] - Octahedron\n";
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" [4] - Dodecahedron\n";
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" [5] - Icosahedron\n";
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(* "[RETURN] - Toggle smooth/flat shading\n"; *) (* not working ... ??? *)
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ignore(Glut.init Sys.argv);
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Glut.initDisplayMode ~alpha:false ~double_buffer:true ~depth:true;
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Glut.initWindowSize ~w:640 ~h:480;
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ignore(Glut.createWindow ~title:"Morph 3D - Shows morphing platonic polyhedra");
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GlClear.color (0.0, 0.0, 0.0);
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GlDraw.color (1.0, 1.0, 1.0);
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GlClear.clear [`color;`depth];
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List.iter ~f:(GlLight.light ~num:0)
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[`ambient ambient; `diffuse diffuse; `position position0];
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List.iter ~f:(GlLight.light ~num:1)
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[`ambient ambient; `diffuse diffuse; `position position1];
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GlLight.light_model (`ambient lmodel_ambient);
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GlLight.light_model (`two_side lmodel_twoside);
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List.iter ~f:Gl.enable
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[`lighting;`light0;`light1;`depth_test;`normalize];
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GlLight.material ~face:`both (`shininess front_shininess);
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GlLight.material ~face:`both (`specular front_specular);
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GlMisc.hint `fog `fastest;
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GlMisc.hint `perspective_correction `fastest;
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GlMisc.hint `polygon_smooth `fastest;
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let view = new view in
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Glut.displayFunc ~cb:(fun () -> view#draw);
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Glut.reshapeFunc ~cb:(fun ~w ~h -> view#reshape w h);
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let rec idle ~value = view#draw; Glut.timerFunc ~ms:20 ~cb:idle ~value:0 in
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Glut.timerFunc ~ms:20 ~cb:idle ~value:0;
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Glut.keyboardFunc ~cb:(fun ~key ~x ~y -> view#keyboard key);
added
removed
LablGlut/examples/lablGL/morph3d.ml
