two new surfaces
- To: mathgroup at smc.vnet.net
- Subject: [mg79084] two new surfaces
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Wed, 18 Jul 2007 02:55:49 -0400 (EDT)
My generalization of the n axial cylinders day before yesterday in OEIS made me think. A little experimentation gave me two new surfaces yesterday morning. Since it started off as a two cylinder and I spelled it wrong, it got a new name ( it's open at only one end!). I call it a two cycle surface: a two cyclinder. It kind of looks like a tulip bud. The torus is the analog of the triaxial torus but you have to have a constant greater than 1 for the hole to be visible. It appears to be a tetrahedral of ellipse that make a torus. For new things you invent, you have to also invent new names. Respectfully, Roger L. Bagula 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocities.com/rlbagulatftn/Index.html alternative email: rlbagula at sbcglobal.net Mathematica: (* Two Cyclinder: not a full torus angular domain*) {x, y, z} = {Cos[p] Cos[t], Sin[p]*Sin[t], Sqrt[(t/Pi)^2 + (p/Pi)^2]} gp=First@ParametricPlot3D[{x,y,z}, {t,0,Pi},{p,0,Pi}, ViewPoint -> {2.8, -1.9, 0.1}, PlotPoints->{20,30}, Boxed->False, Axes -> False, LightSources -> {{{0.7071, 0, 0.7071}, RGBColor[0.9481, 0, 0]}, {{0.5773, 0.5773, 0.5773}, RGBColor[0, 0.8888, 0]}, {{0, 0.7071, 0.7071}, RGBColor[0, 0, 1]}} ]; gp2 = First@ ParametricPlot3D[{x, y, z}, {t, -Pi, 0}, {p, 0, Pi}, ViewPoint -> { 2.8, -1.9, 0.1}, PlotPoints -> {20, 30}, Boxed -> False, Axes -> False, LightSources -> {{{0.7071, 0, 0.7071}, RGBColor[0.9481, 0, 0]}, {{0.5773, 0.5773, 0.5773}, RGBColor[0, 0.8888, 0]}, {{0, 0.7071, 0.7071}, RGBColor[0, 0, 1]}}]; Show[Graphics3D[{gp /. Polygon -> Line, EdgeForm[GrayLevel[.8]], gp2}, {Boxed -> False, ViewPoint -> {0.832, 2.671, 1.904}}]]; Clear[x,y,x,gp,gp2] (* two torus: angular domain is different*) (* http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/2torus/ *) {x, y, z} = { Cos[p]*( 2^(1/4) + Cos[t]),Sin[p]*( 2^(1/4) + Sin[t]), Sqrt[(t/Pi)^2 + (p/Pi)^2]} gp=First@ParametricPlot3D[{x,y,z}, {t,0,Pi},{p,-Pi,Pi}, ViewPoint -> {2.8, -1.9, 0.1}, PlotPoints->{20,30}, Boxed->False, Axes -> False, LightSources -> {{{0.7071, 0, 0.7071}, RGBColor[0.9481, 0, 0]}, {{0.5773, 0.5773, 0.5773}, RGBColor[0, 0.8888, 0]}, {{0, 0.7071, 0.7071}, RGBColor[0, 0, 1]}} ]; gp2 = First@ ParametricPlot3D[{x, y, z}, {t, -Pi, 0}, {p, - Pi, Pi}, ViewPoint -> {2.8, -1.9, 0.1}, PlotPoints -> {20, 30}, Boxed -> False, Axes -> False, LightSources -> {{{0.7071, 0, 0.7071}, RGBColor[0.9481, 0, 0]}, {{ 0.5773, 0.5773, 0.5773}, RGBColor[0, 0.8888, 0]}, {{0, 0.7071, 0.7071}, RGBColor[0, 0, 1]}}]; Show[Graphics3D[{gp /. Polygon -> Line, EdgeForm[GrayLevel[.8]], gp2}, {Boxed -> False, ViewPoint -> {0.832, 2.671, 1.904}}]]; Show[Graphics3D[{gp /. Polygon -> Line, EdgeForm[GrayLevel[.8]], gp2}, {Boxed -> False, ViewPoint -> {-3.364, - 0.164, 0.329}}]];