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Diffstat (limited to 'buch/chapters/030-geometrie/images/kegelschnitte.pov')
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diff --git a/buch/chapters/030-geometrie/images/kegelschnitte.pov b/buch/chapters/030-geometrie/images/kegelschnitte.pov new file mode 100644 index 0000000..60e6964 --- /dev/null +++ b/buch/chapters/030-geometrie/images/kegelschnitte.pov @@ -0,0 +1,492 @@ +// +// kegelschnitte.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.090; +#declare r = 0.03; +#declare R = 1.3 * r; + +camera { + location <-33, 20, 50> + look_at <0, -1.30, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <10, 5, 40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from <from> to <to> with thickness <arrowthickness> with +// color <c> +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +//arrow(<-5,0,0>, <5,0,0>, r, White) +//arrow(<0,-2,0>, <0,2,0>, r, White) +//arrow(<0,0,-2>, <0,0,2>, r, White) + +#declare kegelfarbe = rgb<0.2,0.6,0.6>; +#declare kegelfarbetransparent = rgbt<0.6,0.6,0.6,0.7>; +#declare fokusfarbe = rgb<1,0.2,0.2>; +#declare scheitelfarbe = rgb<0.0,0.6,0.0>; +#declare kurvenfarbe = rgb<0.4,0.8,0>; +#declare leitfarbe = rgb<0.8,0.2,0.8>; +#declare ebenenfarbe = rgbt<0.6,0.4,0.2,0.0>; + +#declare kegelhoehe = 3; +#declare kegelradius = 1.4; + +#macro kegel(X) +union { + cone { <X,0,0>, 0, <X,-kegelhoehe,0>, kegelradius } + cone { <X,0,0>, 0, <X,kegelhoehe,0>, kegelradius } +} +#end + +#macro ebene(X, normale, abstand, anteil) +difference { + intersection { + box { + <X - anteil*(kegelradius + 0.00), -kegelhoehe, 10>, + //<X - 0.2, -kegelhoehe, 10>, + <X + 0.9*(kegelradius + 0.00), kegelhoehe, -10> + } + plane { normale, abstand + 0.001 } + plane { -normale, -abstand + 0.001 } + } + kegel(X) + no_shadow + pigment { + color ebenenfarbe + } + finish { + specular 0.5 + metallic + } +} +#end + +#declare nparabel = vnormalize(<0, kegelradius, kegelhoehe>); +#declare nparabel2 = vnormalize(<0, kegelradius, -kegelhoehe>); +#declare nellipse = vnormalize(<0, kegelradius + 2, kegelhoehe>); +#declare nellipse2 = vnormalize(<0, kegelradius + 2, -kegelhoehe>); +#declare nhyperbel = vnormalize(<0, kegelradius - 1, kegelhoehe>); + +#declare offsetparabel = -0.6; +#declare offsetellipse = -1.1; +#declare offsethyperbel = 0.4; + +// +// Hyperbel +// +ebene(3, nhyperbel, offsethyperbel, 0.0) +intersection { + plane { nhyperbel, offsethyperbel } + kegel(3) + pigment { + color kegelfarbe + } + finish { + specular 0.95 + metallic + } +} + +intersection { + plane { -nhyperbel, -offsethyperbel-0.001 } + kegel(3) + no_shadow + pigment { + color kegelfarbetransparent + } + finish { + specular 0.5 + metallic + } +} + + +#declare e3 = <0, 1, 0>; +#declare tunten = -offsethyperbel / vdot(nparabel-nhyperbel, e3); +#declare toben = offsethyperbel / vdot(nparabel+nhyperbel, e3); +#declare winkel = acos(vdot(nparabel, nhyperbel)); + +#declare Cunten = e3 * tunten + <3,0,0>; +#declare Coben = e3 * toben + <3,0,0>; + +#declare runten = -(vdot(Cunten, nhyperbel) - offsethyperbel); +#declare Funten = Cunten + runten * nhyperbel; +#declare Punten = Cunten + runten * nparabel; +#declare Sunten = Cunten + runten * vnormalize(nparabel+nhyperbel) / cos(winkel/2); + +#declare roben = -(vdot(Coben, nhyperbel) - offsethyperbel); +#declare Foben = Coben + roben * nhyperbel; +#declare Poben = Coben - roben * nparabel2; +#declare Soben = Coben + roben * vnormalize(-nparabel2+nhyperbel) / cos(winkel/2); + +#declare Origin = (Soben + Sunten) / 2; +#declare xaxis = vnormalize(Sunten - Origin); +#declare yaxis = <1,0,0>; + +#declare a = vlength(Sunten - Soben) / 2; +#declare e = vlength(Foben - Funten) / 2; +#declare b = sqrt(e*e - a*a); + +#macro hyperbel(s) + Origin + (a * cosh(s) * xaxis + b * sinh(s) * yaxis) +#end + +#macro hyperbelneg(s) + Origin + (-a * cosh(s) * xaxis + b * sinh(s) * yaxis) +#end + +#declare rkurve = 0.6 * r; + +// Hyperbel +union { + #declare N = 100; + #declare smin = -1.72; + #declare smax = -smin; + #declare sstep = (smax - smin) / N; + #declare s = smin; + #while (s < smax - sstep/2) + cylinder { hyperbel(s), hyperbel(s+sstep), rkurve } + cylinder { hyperbelneg(s), hyperbelneg(s+sstep), rkurve } + sphere { hyperbel(s), rkurve } + sphere { hyperbelneg(s), rkurve } + #declare s = s + sstep; + #end + sphere { hyperbel(s), rkurve } + sphere { hyperbelneg(s), rkurve } + pigment { + color kurvenfarbe + } + finish { + specular 0.9 + metallic + } +} + +// Brennpunkte +union { + sphere { Funten, r } + sphere { Foben, r } + pigment { + color fokusfarbe + } + finish { + specular 0.9 + metallic + } +} + +// Scheitelpunkte +union { + sphere { Sunten, r } + sphere { Soben, r } + pigment { + color scheitelfarbe + } + finish { + specular 0.9 + metallic + } +} + +#declare swert = -1.3; +union { + sphere { hyperbel(swert), r } + cylinder { Funten, hyperbel(swert), rkurve } + cylinder { Foben, hyperbel(swert), rkurve } + pigment { + color Yellow + } + finish { + specular 0.9 + metallic + } +} +// +// Parabeln +// +intersection { + plane { nparabel, offsetparabel } + kegel(0) + pigment { + color kegelfarbe + } + finish { + specular 0.95 + metallic + } +} + +intersection { + plane { -nparabel, -offsetparabel+0.001 } + kegel(0) + //no_shadow + pigment { + color kegelfarbetransparent + } + finish { + specular 0.5 + metallic + } +} + +ebene(0, nparabel, offsetparabel, 0.5) + +#declare tcenter = 0.5 * offsetparabel / vdot(nparabel, e3); +#declare C = tcenter * e3; +#declare F = C + 0.5 * offsetparabel * nparabel; +#declare S = <0, tcenter, tcenter * (kegelradius/kegelhoehe) >; +#declare xaxis = vnormalize(F - S); +#declare p = 4 * vlength(S - F); + +#declare L = S + (S-F); + +#macro parabel(Y) + (S + Y * yaxis + Y * (Y/p) * xaxis) +#end +#macro leitgerade(Y) + L + Y * yaxis +#end + +// +// Parabel +// +union { + #declare ymin = -1.19; + #declare ymax = -ymin; + #declare ystep = (ymax - ymin) / N; + #declare Y = ymin; + #while (Y < ymax - ystep/2) + sphere { parabel(Y), rkurve } + cylinder { parabel(Y), parabel(Y+ystep), rkurve } + #declare Y = Y + ystep; + #end + sphere { parabel(Y), rkurve } + pigment { + color kurvenfarbe + } + finish { + specular 0.9 + metallic + } +} + +union { + cylinder { leitgerade(ymin), leitgerade(ymax), rkurve } + sphere { leitgerade(ymin), rkurve} + sphere { leitgerade(ymax), rkurve} + pigment { + color leitfarbe + } + finish { + specular 0.9 + metallic + } +} + +union { + sphere { F, r } + pigment { + color fokusfarbe + } + finish { + specular 0.9 + metallic + } +} + +union { + sphere { S, r } + pigment { + color scheitelfarbe + } + finish { + specular 0.9 + metallic + } +} + +#declare ywert = -0.7; +union { + sphere { parabel(ywert), r } + cylinder { F, parabel(ywert), rkurve } + cylinder { parabel(ywert), leitgerade(ywert), rkurve } + sphere { leitgerade(ywert), r } + pigment { + color Yellow + } + finish { + specular 0.9 + metallic + } +} + +// +// Dandelin Kugeln für die Ellipse +// + +ebene(-3, nellipse, offsetellipse, 1) +intersection { + plane { nellipse, offsetellipse } + kegel(-3) + pigment { + color kegelfarbe + } + finish { + specular 0.95 + metallic + } +} + +intersection { + plane { -nellipse, -offsetellipse+0.001 } + kegel(-3) + //no_shadow + pigment { + color kegelfarbetransparent + } + finish { + specular 0.5 + metallic + } +} + +#declare tunten = -offsetellipse / vdot(nparabel-nellipse, e3); +#declare toben = offsetellipse / vdot(nparabel+nellipse, e3); +#declare winkel = acos(vdot(nparabel, nellipse)); +#declare winkel2 = acos(-vdot(nparabel2, nellipse)); + +#declare Cunten = e3 * tunten + <-3,0,0>; +#declare Coben = e3 * toben + <-3,0,0>; + +#declare runten = -(vdot(Cunten, nellipse) - offsetellipse); +#declare Funten = Cunten + runten * nellipse; +#declare Punten = Cunten + runten * nparabel; +#declare Sunten = Cunten + runten * vnormalize(nparabel+nellipse) / cos(winkel/2); + +#declare roben = (vdot(Coben, nellipse) - offsetellipse); +#declare Foben = Coben - roben * nellipse; +#declare Poben = Coben + roben * nparabel2; +#declare Soben = Coben - roben * vnormalize(-nparabel2+nellipse) / cos(winkel2/2); + +#declare Origin = (Soben + Sunten) / 2; +#declare xaxis = vnormalize(Sunten - Origin); +#declare yaxis = <1,0,0>; + +#declare a = vlength(Sunten - Soben) / 2; +#declare e = vlength(Foben - Funten) / 2; +#declare b = sqrt(a*a-e*e); + +#macro ellipse(s) + Origin + (a * cos(s) * xaxis + b * sin(s) * yaxis) +#end + +union { + #declare N = 100; + #declare smin = -pi; + #declare smax = -smin; + #declare sstep = (smax - smin) / N; + #declare s = smin; + #while (s < smax - sstep/2) + cylinder { ellipse(s), ellipse(s+sstep), rkurve } + sphere { ellipse(s), rkurve } + #declare s = s + sstep; + #end + sphere { ellipse(s), rkurve } + pigment { + color kurvenfarbe + } + finish { + specular 0.9 + metallic + } +} + +union { + //sphere { Cunten, runten + 0.001 } + //sphere { Coben, roben + 0.001 } + sphere { Funten, r } + sphere { Foben, r } + pigment { + color fokusfarbe + } + finish { + specular 0.9 + metallic + } +} + +union { + sphere { Sunten, r } + sphere { Soben, r } + pigment { + color scheitelfarbe + } + finish { + specular 0.9 + metallic + } +} + +#declare swert = -1.2; +union { + sphere { ellipse(swert), r } + cylinder { Funten, ellipse(swert), rkurve } + cylinder { Foben, ellipse(swert), rkurve } + pigment { + color Yellow + } + finish { + specular 0.9 + metallic + } +} |