static void plot_curve_accurate(const Curve &curve, float step, T plot_func) {
if (curve.get_point_count() <= 1) {
// Not enough points to make a curve, so it's just a straight line
float y = curve.interpolate(0);
plot_func(Vector2(0, y), Vector2(1.f, y), true);
} else {
Vector2 first_point = curve.get_point_position(0);
Vector2 last_point = curve.get_point_position(curve.get_point_count() - 1);
// Edge lines
plot_func(Vector2(0, first_point.y), first_point, false);
plot_func(Vector2(Curve::MAX_X, last_point.y), last_point, false);
// Draw section by section, so that we get maximum precision near points.
// It's an accurate representation, but slower than using the baked one.
for (int i = 1; i < curve.get_point_count(); ++i) {
Vector2 a = curve.get_point_position(i - 1);
Vector2 b = curve.get_point_position(i);
Vector2 pos = a;
Vector2 prev_pos = a;
float len = b.x - a.x;
for (float x = step; x < len; x += step) {
pos.x = a.x + x;
pos.y = curve.interpolate_local_nocheck(i - 1, x);
plot_func(prev_pos, pos, true);
prev_pos = pos;
}
plot_func(prev_pos, b, true);
}
}
}
main()
{
double ans, exact, err;
int m;
double eps = 1.0e-6;
double a = 0.0;
double b = PI;
int n_jac = 12;
std::vector<double> x_jac(n_jac);
std::vector<double> w_jac(n_jac);
gauss_jacobi(x_jac, w_jac, n_jac, 1.5, -0.5);
printf("\n\n Abscissas and wights from gauss_jacobi with alpha = 1.5, beta = -0.5.\n");
for (int i = 0; i < n_jac; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_jac[i], i, w_jac[i]);
int n_leg = 12;
std::vector<double> x_leg(n_leg);
std::vector<double> w_leg(n_leg);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
printf("\n\n Abscissas and wights from gauss_legendre.\n");
for (int i = 0; i < n_leg; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_leg[i] / PIO2 - 1.0, i, w_leg[i] / PIO2);
int n_jac2 = 12;
std::vector<double> x_jac2(n_jac2);
std::vector<double> w_jac2(n_jac2);
gauss_jacobi(x_jac2, w_jac2, n_jac2, 0.0, 0.0);
printf("\n\n Abscissas and wights from gauss_jacobi with alpha = beta = 0.0.\n");
for (int i = 0; i < n_jac2; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_jac2[i], i, w_jac2[i]);
int n_cheb = 12;
std::vector<double> x_cheb(n_cheb);
std::vector<double> w_cheb(n_cheb);
gauss_chebyshev(x_cheb, w_cheb, n_cheb);
printf("\n\n Abscissas and wights from gauss_chebyshev.\n");
for (int i = 0; i < (n_cheb + 1) / 2; ++i)
printf("\n x[%d] = -x[%d] = %16.12f w[%d] = w[%d] = %16.12f",
i, n_cheb + 1 - i , x_cheb[i], i, n_cheb + 1 - i, w_cheb[i]);
int n_jac3 = 12;
std::vector<double> x_jac3(n_jac3);
std::vector<double> w_jac3(n_jac3);
gauss_jacobi(x_jac3, w_jac3, n_jac3, -0.5, -0.5);
printf("\n\n Abscissas and wights from gauss_jacobi with alpha = beta = -0.5.\n");
for (int i = 0; i < n_jac3; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i, x_jac3[i], i, w_jac3[i]);
int n_herm = 12;
std::vector<double> x_herm(n_herm);
std::vector<double> w_herm(n_herm);
gauss_hermite(x_herm, w_herm, n_herm);
printf("\n\n Abscissas and wights from gauss_hermite.\n");
for (int i = 0; i < (n_herm + 1) / 2; ++i)
printf("\n x[%d] = -x[%d] = %16.12f w[%d] = w[%d] = %16.12f",
i, n_herm + 1 - i , x_herm[i], i, n_herm + 1 - i, w_herm[i]);
n_lag = 12;
std::vector<double> x_lag(n_lag);
std::vector<double> w_lag(n_lag);
gauss_laguerre(x_lag, w_lag, n_lag, 1.0);
printf("\n\n Abscissas and wights from gauss_laguerre.\n");
for (int i = 0; i < n_lag; ++i)
printf("\n x[%d] = %16.12f w[%d] = %16.12f",
i , x_lag[i], i, w_lag[i]);
m = 40;
std::vector<double> c(m);
std::vector<double> cint(m);
std::vector<double> cder(m);
printf("\n\n\n\n Test of integration routines...");
printf("\n\n");
printf("\n\n %-40s %g", "Input requested error", eps);
printf("\n\n %-40s %d", "Input order of Gaussian quadrature", n_leg);
printf("\n\n %-40s %d", "Input order of Chebyshev fit", m);
printf("\n\n");
a = 0.0;
b = PI;
printf("\n\n Integrate cos(x) from a = %f to b = %f . . .", a, PI);
exact = 0.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(std::cos, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(std::cos, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg(std::cos, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(std::cos, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, cos);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate sin(x) from a = %f to b = %f . . .", a, b);
exact = 2.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(sin, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(sin, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg(sin, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(sin, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, sin);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate cos^2(x) from a = %f to b = %f . . .", a, b);
exact = PI/2.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(cos2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(cos2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg(cos2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(cos2, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, cos2);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate sin^2(x) from a = %f to b = %f . . .", a, b);
exact = PI/2.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(sin2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(sin2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's integral", ans, err);
ans = quad_romberg(sin2, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(sin2, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, sin2);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate J_1(x) from a = %f to b = %f . . .", a, b);
exact = bessel_j0(0.0) - bessel_j0(PI);
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(bessel_j1, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(bessel_j1, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's integral", ans, err);
ans = quad_romberg(bessel_j1, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(bessel_j1, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
chebyshev_fit(a, b, c, m, bessel_j1);
chebyshev_integ(a, b, c, cint, m);
chebyshev_deriv(a, b, c, cder, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = 10.0*PI;
printf("\n\n Integrate foo(x) = (1 - x)exp(-x/2) from a = %f to b = %f . . .", a, b);
exact = 2.0*(1.0 + b)*exp(-b/2.0) - 2.0*(1.0 + a)*exp(-a/2.0);
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_trapezoid(foo, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Trapezoid rule", ans, err);
ans = quad_simpson(foo, a, b, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Simpson's rule", ans, err);
ans = quad_romberg_open(foo, a, b, eps, midpoint_exp);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg integration", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(foo, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
gauss_laguerre(x_lag, w_lag, n_lag, 0.0);
ans = 0.0;
for (int i = 0; i < n_lag; ++i)
ans += 2 * w_lag[i] * foonum(2.0 * x_lag[i]);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Laguerre quadrature", ans, err);
chebyshev_fit(a, b, c, m, foo);
chebyshev_integ(a, b, c, cint, m);
chebyshev_deriv(a, b, c, cder, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = clenshaw_curtis_quad(a, b, c, m, eps);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Clenshaw - Curtis quadrature", ans, err);
a = 0.0;
b = PI;
printf("\n\n Integrate funk1(x) = cos(x)/sqrt(x(PI - x)) from a = %f to b = %f . . .", a, b);
exact = 0.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_romberg_open(funk1, a, b, eps, midpoint);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg quadrature with midpoint", ans, err);
ans = quad_romberg_open(funk1, a, (a+b)/2, eps, midpoint_inv_sqrt_lower)
+ quad_romberg_open(funk1, (a+b)/2, b, eps, midpoint_inv_sqrt_upper);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg with inverse sqrt step", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(funk1, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
ans = quad_gauss(funk1num, a, b, x_cheb, w_cheb, n_cheb);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Chebyshev quadrature", ans, err);
ans = quad_gauss(funk1num, a, b, x_jac, w_jac, n_jac);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Jacobi quadrature", ans, err);
chebyshev_fit(a, b, c, m, funk1);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = dumb_gauss_crap(funk1num, a, b, 8);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Chebyshev quadrature", ans, err);
printf("\n\n");
plot_func(funk1, a+0.1, b-0.1, "", "", "", "");
a = 0.0;
b = PI;
printf("\n\n Integrate funk2(x) = (2.0+sin(x))/sqrt(x(PI - x)) from a = %f to b = %f . . .", a, b);
exact = 0.0;
printf("\n %-40s %16.12f", "Exact answer", exact);
ans = quad_romberg_open(funk2, a, b, eps, midpoint);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg quadrature with midpoint", ans, err);
ans = quad_romberg_open(funk2, a, (a+b)/2, eps, midpoint_inv_sqrt_lower)
+ quad_romberg_open(funk2, (a+b)/2, b, eps, midpoint_inv_sqrt_upper);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Open Romberg with inverse sqrt step", ans, err);
gauss_legendre(n_leg, a, b, x_leg, w_leg);
ans = quad_gauss_legendre(funk2, x_leg, w_leg, n_leg);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Legendre quadrature", ans, err);
ans = quad_gauss(funk2num, a, b, x_cheb, w_cheb, n_cheb);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Chebyshev quadrature", ans, err);
ans = quad_gauss(funk2num, a, b, x_jac, w_jac, n_jac);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Gauss - Jacobi quadrature", ans, err);
chebyshev_fit(a, b, c, m, funk2);
chebyshev_integ(a, b, c, cint, m);
ans = chebyshev_eval(a, b, cint, m, b);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Chebyshev evaluation of integral", ans, err);
ans = dumb_gauss_crap(funk2num, a, b, 8);
err = ans - exact;
printf("\n %-40s %16.12f %16.12f", "Adaptive Gauss - Chebyshev quadrature", ans, err);
printf("\n\n");
plot_func(funk2, a+0.1, b-0.1, "", "", "", "");
printf("\n\n");
}
void CurveEditor::_draw() {
if (_curve_ref.is_null())
return;
Curve &curve = **_curve_ref;
update_view_transform();
// Background
Vector2 view_size = get_rect().size;
draw_style_box(get_stylebox("bg", "Tree"), Rect2(Point2(), view_size));
// Grid
draw_set_transform_matrix(_world_to_view);
Vector2 min_edge = get_world_pos(Vector2(0, view_size.y));
Vector2 max_edge = get_world_pos(Vector2(view_size.x, 0));
const Color grid_color0 = get_color("grid_major_color", "Editor");
const Color grid_color1 = get_color("grid_minor_color", "Editor");
draw_line(Vector2(min_edge.x, curve.get_min_value()), Vector2(max_edge.x, curve.get_min_value()), grid_color0);
draw_line(Vector2(max_edge.x, curve.get_max_value()), Vector2(min_edge.x, curve.get_max_value()), grid_color0);
draw_line(Vector2(0, min_edge.y), Vector2(0, max_edge.y), grid_color0);
draw_line(Vector2(1, max_edge.y), Vector2(1, min_edge.y), grid_color0);
float curve_height = (curve.get_max_value() - curve.get_min_value());
const Vector2 grid_step(0.25, 0.5 * curve_height);
for (real_t x = 0; x < 1.0; x += grid_step.x) {
draw_line(Vector2(x, min_edge.y), Vector2(x, max_edge.y), grid_color1);
}
for (real_t y = curve.get_min_value(); y < curve.get_max_value(); y += grid_step.y) {
draw_line(Vector2(min_edge.x, y), Vector2(max_edge.x, y), grid_color1);
}
// Markings
draw_set_transform_matrix(Transform2D());
Ref<Font> font = get_font("font", "Label");
float font_height = font->get_height();
Color text_color = get_color("font_color", "Editor");
{
// X axis
float y = curve.get_min_value();
Vector2 off(0, font_height - 1);
draw_string(font, get_view_pos(Vector2(0, y)) + off, "0.0", text_color);
draw_string(font, get_view_pos(Vector2(0.25, y)) + off, "0.25", text_color);
draw_string(font, get_view_pos(Vector2(0.5, y)) + off, "0.5", text_color);
draw_string(font, get_view_pos(Vector2(0.75, y)) + off, "0.75", text_color);
draw_string(font, get_view_pos(Vector2(1, y)) + off, "1.0", text_color);
}
{
// Y axis
float m0 = curve.get_min_value();
float m1 = 0.5 * (curve.get_min_value() + curve.get_max_value());
float m2 = curve.get_max_value();
Vector2 off(1, -1);
draw_string(font, get_view_pos(Vector2(0, m0)) + off, String::num(m0, 2), text_color);
draw_string(font, get_view_pos(Vector2(0, m1)) + off, String::num(m1, 2), text_color);
draw_string(font, get_view_pos(Vector2(0, m2)) + off, String::num(m2, 3), text_color);
}
// Draw tangents for current point
if (_selected_point >= 0) {
const Color tangent_color = get_color("accent_color", "Editor");
int i = _selected_point;
Vector2 pos = curve.get_point_position(i);
if (i != 0) {
Vector2 control_pos = get_tangent_view_pos(i, TANGENT_LEFT);
draw_line(get_view_pos(pos), control_pos, tangent_color);
draw_rect(Rect2(control_pos, Vector2(1, 1)).grow(2), tangent_color);
}
if (i != curve.get_point_count() - 1) {
Vector2 control_pos = get_tangent_view_pos(i, TANGENT_RIGHT);
draw_line(get_view_pos(pos), control_pos, tangent_color);
draw_rect(Rect2(control_pos, Vector2(1, 1)).grow(2), tangent_color);
}
}
// Draw lines
draw_set_transform_matrix(_world_to_view);
const Color line_color = get_color("highlight_color", "Editor");
const Color edge_line_color = get_color("font_color", "Editor");
CanvasItemPlotCurve plot_func(*this, line_color, edge_line_color);
plot_curve_accurate(curve, 4.f / view_size.x, plot_func);
// Draw points
draw_set_transform_matrix(Transform2D());
const Color point_color = get_color("font_color", "Editor");
const Color selected_point_color = get_color("accent_color", "Editor");
for (int i = 0; i < curve.get_point_count(); ++i) {
Vector2 pos = curve.get_point_position(i);
draw_rect(Rect2(get_view_pos(pos), Vector2(1, 1)).grow(3), i == _selected_point ? selected_point_color : point_color);
// TODO Circles are prettier. Needs a fix! Or a texture
//draw_circle(pos, 2, point_color);
}
// Hover
if (_hover_point != -1) {
const Color hover_color = line_color;
Vector2 pos = curve.get_point_position(_hover_point);
stroke_rect(Rect2(get_view_pos(pos), Vector2(1, 1)).grow(_hover_radius), hover_color);
}
// Help text
if (_selected_point > 0 && _selected_point + 1 < curve.get_point_count()) {
text_color.a *= 0.4;
draw_string(font, Vector2(50, font_height), TTR("Hold Shift to edit tangents individually"), text_color);
}
}
static void draw_func_name (FILE *fp, int x, int index)
{
char *string = plot_func (index);
fprintf (fp, "(%s) %d %d name\n", string, x, 0);
}
