From e8bb3fd399f2261c9b430ffa319626950499d4c1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 23 May 2022 23:06:11 +0200 Subject: new spherical graph --- buch/papers/kugel/images/Makefile | 12 ++- buch/papers/kugel/images/spherecurve.m | 160 ++++++++++++++++++++++++++++ buch/papers/kugel/images/spherecurve.maxima | 13 +++ buch/papers/kugel/images/spherecurve.pov | 73 +++++++++++++ 4 files changed, 257 insertions(+), 1 deletion(-) create mode 100644 buch/papers/kugel/images/spherecurve.m create mode 100644 buch/papers/kugel/images/spherecurve.maxima create mode 100644 buch/papers/kugel/images/spherecurve.pov (limited to 'buch/papers') diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile index e8bf919..6187fed 100644 --- a/buch/papers/kugel/images/Makefile +++ b/buch/papers/kugel/images/Makefile @@ -3,7 +3,7 @@ # # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: curvature.jpg +all: curvature.jpg spherecurve.jpg curvature.inc: curvgraph.m octave curvgraph.m @@ -13,3 +13,13 @@ curvature.png: curvature.pov curvature.inc curvature.jpg: curvature.png convert curvature.png -density 300 -units PixelsPerInch curvature.jpg + +spherecurve.inc: spherecurve.m + octave spherecurve.m + +spherecurve.png: spherecurve.pov spherecurve.inc + povray +A0.1 +W1920 +H1080 +Ospherecurve.png spherecurve.pov + +spherecurve.jpg: spherecurve.png + convert spherecurve.png -density 300 -units PixelsPerInch spherecurve.jpg + diff --git a/buch/papers/kugel/images/spherecurve.m b/buch/papers/kugel/images/spherecurve.m new file mode 100644 index 0000000..ea9c901 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.m @@ -0,0 +1,160 @@ +# +# spherecurv.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global a; +a = 5; +global A; +A = 10; + +phisteps = 400; +hphi = 2 * pi / phisteps; +thetasteps = 200; +htheta = pi / thetasteps; + +function retval = f(z) + global a; + global A; + retval = A * exp(a * (z^2 - 1)); +endfunction + +function retval = g(z) + global a; + retval = -f(z) * 2 * a * (2 * a * z^4 + (3 - 2*a) * z^2 - 1); + # 2 + # - a 2 4 2 2 a z + #(%o6) - %e (4 a z + (6 a - 4 a ) z - 2 a) %e +endfunction + +phi = (1 + sqrt(5)) / 2; + +global axes; +axes = [ + 0, 0, 1, -1, phi, -phi; + 1, -1, phi, phi, 0, 0; + phi, phi, 0, 0, 1, 1; +]; +axes = axes / (sqrt(phi^2+1)); + +function retval = kugel(theta, phi) + retval = [ + cos(phi) * sin(theta); + sin(phi) * sin(theta); + cos(theta) + ]; +endfunction + +function retval = F(v) + global axes; + s = 0; + for i = (1:6) + z = axes(:,i)' * v; + s = s + f(z); + endfor + retval = s / 6; +endfunction + +function retval = F2(theta, phi) + v = kugel(theta, phi); + retval = F(v); +endfunction + +function retval = G(v) + global axes; + s = 0; + for i = (1:6) + s = s + g(axes(:,i)' * v); + endfor + retval = s / 6; +endfunction + +function retval = G2(theta, phi) + v = kugel(theta, phi); + retval = G(v); +endfunction + +function retval = cnormalize(u) + utop = 11; + ubottom = -30; + retval = (u - ubottom) / (utop - ubottom); + if (retval > 1) + retval = 1; + endif + if (retval < 0) + retval = 0; + endif +endfunction + +global umin; +umin = 0; +global umax; +umax = 0; + +function color = farbe(v) + global umin; + global umax; + u = G(v); + if (u < umin) + umin = u; + endif + if (u > umax) + umax = u; + endif + u = cnormalize(u); + color = [ u, 0.5, 1-u ]; + color = color/max(color); +endfunction + +function dreieck(fn, v0, v1, v2) + fprintf(fn, " mesh {\n"); + c = (v0 + v1 + v2) / 3; + c = c / norm(c); + color = farbe(c); + v0 = v0 * (1 + F(v0)); + v1 = v1 * (1 + F(v1)); + v2 = v2 * (1 + F(v2)); + fprintf(fn, "\ttriangle {\n"); + fprintf(fn, "\t <%.6f,%.6f,%.6f>,\n", v0(1,1), v0(3,1), v0(2,1)); + fprintf(fn, "\t <%.6f,%.6f,%.6f>,\n", v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "\t <%.6f,%.6f,%.6f>\n", v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "\t}\n"); + fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n", + color(1,1), color(1,2), color(1,3)); + fprintf(fn, "\tfinish { metallic specular 0.5 }\n"); + fprintf(fn, " }\n"); +endfunction + +fn = fopen("spherecurve.inc", "w"); + + for i = (1:phisteps) + # Polkappe nord + v0 = [ 0; 0; 1 ]; + v1 = kugel(htheta, (i-1) * hphi); + v2 = kugel(htheta, i * hphi); + fprintf(fn, " // i = %d\n", i); + dreieck(fn, v0, v1, v2); + + # Polkappe sued + v0 = [ 0; 0; -1 ]; + v1 = kugel(pi-htheta, (i-1) * hphi); + v2 = kugel(pi-htheta, i * hphi); + dreieck(fn, v0, v1, v2); + endfor + + for j = (1:thetasteps-2) + for i = (1:phisteps) + v0 = kugel( j * htheta, (i-1) * hphi); + v1 = kugel((j+1) * htheta, (i-1) * hphi); + v2 = kugel( j * htheta, i * hphi); + v3 = kugel((j+1) * htheta, i * hphi); + fprintf(fn, " // i = %d, j = %d\n", i, j); + dreieck(fn, v0, v1, v2); + dreieck(fn, v1, v2, v3); + endfor + endfor + +fclose(fn); + +umin +umax diff --git a/buch/papers/kugel/images/spherecurve.maxima b/buch/papers/kugel/images/spherecurve.maxima new file mode 100644 index 0000000..1e9077c --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.maxima @@ -0,0 +1,13 @@ +/* + * spherecurv.maxima + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +f: exp(-a * sin(theta)^2); + +g: ratsimp(diff(sin(theta) * diff(f, theta), theta)/sin(theta)); +g: subst(z, cos(theta), g); +g: subst(sqrt(1-z^2), sin(theta), g); +ratsimp(g); + +f: ratsimp(subst(sqrt(1-z^2), sin(theta), f)); diff --git a/buch/papers/kugel/images/spherecurve.pov b/buch/papers/kugel/images/spherecurve.pov new file mode 100644 index 0000000..86c3745 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.pov @@ -0,0 +1,73 @@ +// +// curvature.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.14; + +camera { + location <10, 10, -40> + look_at <0, 0, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-10, 10, -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 to with thickness with +// color +// +#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(<-2.7,0,0>, <2.7,0,0>, 0.03, White) +arrow(<0,-2.7,0>, <0,2.7,0>, 0.03, White) +arrow(<0,0,-2.7>, <0,0,2.7>, 0.03, White) + +#include "spherecurve.inc" + -- cgit v1.2.1 From 6ee6a7b0cf91469c7a79827293b8e3b880a6a0aa Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 24 May 2022 11:45:50 +0200 Subject: add C++ program to compute the surface --- buch/papers/kugel/images/Makefile | 9 +- buch/papers/kugel/images/spherecurve.cpp | 292 +++++++++++++++++++++++++++++++ buch/papers/kugel/images/spherecurve.m | 4 +- buch/papers/kugel/images/spherecurve.pov | 4 +- 4 files changed, 303 insertions(+), 6 deletions(-) create mode 100644 buch/papers/kugel/images/spherecurve.cpp (limited to 'buch/papers') diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile index 6187fed..4226dab 100644 --- a/buch/papers/kugel/images/Makefile +++ b/buch/papers/kugel/images/Makefile @@ -14,12 +14,17 @@ curvature.png: curvature.pov curvature.inc curvature.jpg: curvature.png convert curvature.png -density 300 -units PixelsPerInch curvature.jpg -spherecurve.inc: spherecurve.m +spherecurve2.inc: spherecurve.m octave spherecurve.m spherecurve.png: spherecurve.pov spherecurve.inc - povray +A0.1 +W1920 +H1080 +Ospherecurve.png spherecurve.pov + povray +A0.1 +W1080 +H1080 +Ospherecurve.png spherecurve.pov spherecurve.jpg: spherecurve.png convert spherecurve.png -density 300 -units PixelsPerInch spherecurve.jpg +spherecurve: spherecurve.cpp + g++ -o spherecurve -g -Wall -O spherecurve.cpp + +spherecurve.inc: spherecurve + ./spherecurve diff --git a/buch/papers/kugel/images/spherecurve.cpp b/buch/papers/kugel/images/spherecurve.cpp new file mode 100644 index 0000000..eff8c33 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.cpp @@ -0,0 +1,292 @@ +/* + * spherecurve.cpp + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +#include +#include +#include +#include +#include + +inline double sqr(double x) { return x * x; } + +/** + * \brief Class for 3d vectors (also used as colors) + */ +class vector { + double X[3]; +public: + vector() { X[0] = X[1] = X[2] = 0; } + vector(double a) { X[0] = X[1] = X[2] = a; } + vector(double x, double y, double z) { + X[0] = x; X[1] = y; X[2] = z; + } + vector(double theta, double phi) { + double s = sin(theta); + X[0] = cos(phi) * s; + X[1] = sin(phi) * s; + X[2] = cos(theta); + } + vector(const vector& other) { + for (int i = 0; i < 3; i++) { + X[i] = other.X[i]; + } + } + vector operator+(const vector& other) const { + return vector(X[0] + other.X[0], + X[1] + other.X[1], + X[2] + other.X[2]); + } + vector operator*(double l) const { + return vector(X[0] * l, X[1] * l, X[2] * l); + } + double operator*(const vector& other) const { + double s = 0; + for (int i = 0; i < 3; i++) { + s += X[i] * other.X[i]; + } + return s; + } + double norm() const { + double s = 0; + for (int i = 0; i < 3; i++) { + s += sqr(X[i]); + } + return sqrt(s); + } + vector normalize() const { + double l = norm(); + return vector(X[0]/l, X[1]/l, X[2]/l); + } + double max() const { + return std::max(X[0], std::max(X[1], X[2])); + } + double l0norm() const { + double l = 0; + for (int i = 0; i < 3; i++) { + if (fabs(X[i]) > l) { + l = fabs(X[i]); + } + } + return l; + } + vector l0normalize() const { + double l = l0norm(); + vector result(X[0]/l, X[1]/l, X[2]/l); + return result; + } + const double& operator[](int i) const { return X[i]; } + double& operator[](int i) { return X[i]; } +}; + +/** + * \brief Derived 3d vector class implementing color + * + * The constructor in this class converts a single value into a + * color on a suitable gradient. + */ +class color : public vector { +public: + static double utop; + static double ubottom; + static double green; +public: + color(double u) { + u = (u - ubottom) / (utop - ubottom); + if (u > 1) { + u = 1; + } + if (u < 0) { + u = 0; + } + u = pow(u,2); + (*this)[0] = u; + (*this)[1] = green; + (*this)[2] = 1-u; + double l = l0norm(); + for (int i = 0; i < 3; i++) { + (*this)[i] /= l; + } + } +}; + +double color::utop = 12; +double color::ubottom = -31; +double color::green = 0.5; + +/** + * \brief Surface model + * + * This class contains the definitions of the functions to plot + * and the parameters to + */ +class surfacefunction { + static vector axes[6]; + + double _a; + double _A; + + double _umin; + double _umax; +public: + double a() const { return _a; } + double A() const { return _A; } + + double umin() const { return _umin; } + double umax() const { return _umax; } + + surfacefunction(double a, double A) : _a(a), _A(A), _umin(0), _umax(0) { + } + + double f(double z) { + return A() * exp(a() * (sqr(z) - 1)); + } + + double g(double z) { + return -f(z) * 2*a() * ((2*a()*sqr(z) + (3-2*a()))*sqr(z) - 1); + } + + double F(const vector& v) { + double s = 0; + for (int i = 0; i < 6; i++) { + s += f(axes[i] * v); + } + return s / 6; + } + + double G(const vector& v) { + double s = 0; + for (int i = 0; i < 6; i++) { + s += g(axes[i] * v); + } + return s / 6; + } +protected: + color farbe(const vector& v) { + double u = G(v); + if (u < _umin) { + _umin = u; + } + if (u > _umax) { + _umax = u; + } + return color(u); + } +}; + +static double phi = (1 + sqrt(5)) / 2; +static double sl = sqrt(sqr(phi) + 1); +vector surfacefunction::axes[6] = { + vector( 0. , -1./sl, phi/sl ), + vector( 0. , 1./sl, phi/sl ), + vector( 1./sl, phi/sl, 0. ), + vector( -1./sl, phi/sl, 0. ), + vector( phi/sl, 0. , 1./sl ), + vector( -phi/sl, 0. , 1./sl ) +}; + +/** + * \brief Class to construct the plot + */ +class surface : public surfacefunction { + FILE *outfile; + + int _phisteps; + int _thetasteps; + double _hphi; + double _htheta; +public: + int phisteps() const { return _phisteps; } + int thetasteps() const { return _thetasteps; } + double hphi() const { return _hphi; } + double htheta() const { return _htheta; } + void phisteps(int s) { _phisteps = s; _hphi = 2 * M_PI / s; } + void thetasteps(int s) { _thetasteps = s; _htheta = M_PI / s; } + + surface(const std::string& filename, double a, double A) + : surfacefunction(a, A) { + outfile = fopen(filename.c_str(), "w"); + phisteps(400); + thetasteps(200); + } + + ~surface() { + fclose(outfile); + } + +private: + void triangle(const vector& v0, const vector& v1, const vector& v2) { + fprintf(outfile, " mesh {\n"); + vector c = (v0 + v1 + v2) * (1./3.); + vector color = farbe(c.normalize()); + vector V0 = v0 * (1 + F(v0)); + vector V1 = v1 * (1 + F(v1)); + vector V2 = v2 * (1 + F(v2)); + fprintf(outfile, "\ttriangle {\n"); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>,\n", + V0[0], V0[2], V0[1]); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>,\n", + V1[0], V1[2], V1[1]); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>\n", + V2[0], V2[2], V2[1]); + fprintf(outfile, "\t}\n"); + fprintf(outfile, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n", + color[0], color[1], color[2]); + fprintf(outfile, "\tfinish { metallic specular 0.5 }\n"); + fprintf(outfile, " }\n"); + } + + void northcap() { + vector v0(0, 0, 1); + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // northcap i = %d\n", i); + vector v1(htheta(), (i - 1) * hphi()); + vector v2(htheta(), i * hphi()); + triangle(v0, v1, v2); + } + } + + void southcap() { + vector v0(0, 0, -1); + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // southcap i = %d\n", i); + vector v1(M_PI - htheta(), (i - 1) * hphi()); + vector v2(M_PI - htheta(), i * hphi()); + triangle(v0, v1, v2); + } + } + + void zone() { + for (int j = 1; j < thetasteps() - 1; j++) { + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // zone j = %d, i = %d\n", + j, i); + vector v0( j * htheta(), (i-1) * hphi()); + vector v1((j+1) * htheta(), (i-1) * hphi()); + vector v2( j * htheta(), i * hphi()); + vector v3((j+1) * htheta(), i * hphi()); + triangle(v0, v1, v2); + triangle(v1, v2, v3); + } + } + } +public: + void draw() { + northcap(); + southcap(); + zone(); + } +}; + +/** + * \brief main function + */ +int main(int argc, char *argv[]) { + surface S("spherecurve.inc", 5, 10); + color::green = 0.3; + S.draw(); + std::cout << "umin: " << S.umin() << std::endl; + std::cout << "umax: " << S.umax() << std::endl; + return EXIT_SUCCESS; +} diff --git a/buch/papers/kugel/images/spherecurve.m b/buch/papers/kugel/images/spherecurve.m index ea9c901..99d5c9a 100644 --- a/buch/papers/kugel/images/spherecurve.m +++ b/buch/papers/kugel/images/spherecurve.m @@ -1,5 +1,5 @@ # -# spherecurv.m +# spherecurve.m # # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # @@ -125,7 +125,7 @@ function dreieck(fn, v0, v1, v2) fprintf(fn, " }\n"); endfunction -fn = fopen("spherecurve.inc", "w"); +fn = fopen("spherecurve2.inc", "w"); for i = (1:phisteps) # Polkappe nord diff --git a/buch/papers/kugel/images/spherecurve.pov b/buch/papers/kugel/images/spherecurve.pov index 86c3745..b1bf4b8 100644 --- a/buch/papers/kugel/images/spherecurve.pov +++ b/buch/papers/kugel/images/spherecurve.pov @@ -11,12 +11,12 @@ global_settings { assumed_gamma 1 } -#declare imagescale = 0.14; +#declare imagescale = 0.13; camera { location <10, 10, -40> look_at <0, 0, 0> - right 16/9 * x * imagescale + right x * imagescale up y * imagescale } -- cgit v1.2.1 From d08813723e1cab4bca4a527218610023775a4634 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 24 May 2022 11:53:56 +0200 Subject: better color coding --- buch/papers/kugel/images/spherecurve.cpp | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/kugel/images/spherecurve.cpp b/buch/papers/kugel/images/spherecurve.cpp index eff8c33..8ddf5e5 100644 --- a/buch/papers/kugel/images/spherecurve.cpp +++ b/buch/papers/kugel/images/spherecurve.cpp @@ -102,7 +102,7 @@ public: } u = pow(u,2); (*this)[0] = u; - (*this)[1] = green; + (*this)[1] = green * u * (1 - u); (*this)[2] = 1-u; double l = l0norm(); for (int i = 0; i < 3; i++) { @@ -284,7 +284,7 @@ public: */ int main(int argc, char *argv[]) { surface S("spherecurve.inc", 5, 10); - color::green = 0.3; + color::green = 1.0; S.draw(); std::cout << "umin: " << S.umin() << std::endl; std::cout << "umax: " << S.umax() << std::endl; -- cgit v1.2.1