## Sunday, February 3, 2013

### 3D Parametric Curves are cool

3D Parametric Curves are cool because they make it easy to construct beautiful shapes out of simple mathematical formulas. Since I am lazy and not good at modeling 3D scenes, I thought that using these equations was probably the best way to get nice images.

The following curves are made of cylinders, following the paths defined by the parametric curves.

Here are the functions I used to generated the shapes:

void trefoilKnot(float R, float t, float4& p)
{
p.x = R*(sin(t)+2.f*sin(2.f*t));
p.y = R*(cos(t)-2.f*cos(2.f*t));
p.z = R*(-sin(3.f*t));
}

void torus(float R, float t, float4& p )
{
p.x = R*(3.f*cos(t)+cos(10.f*t)*cos(t));
p.y = R*(3.f*sin(t)+cos(10.f*t)*sin(t));
p.z = R*sin(10.f*t);
}

void star(float R, float t, float4& p )
{
p.x = R*(2.f*sin(3.f*t)*cos(t));
p.y = R*(2.f*sin(3.f*t)*sin(t));
p.z = R*sin(3.f*t);
}

void spring(float R, float t, float4& p)
{
p.x = R*cos(t);
p.y = R*sin(t);
p.z = R*cos(t);
}

void heart(float R, float u, float v, float4& p)
{
p.x = R*4.f*pow(sin(u),3.f);
p.y = R*0.25f*(13*cos(u)-5*cos(2.f*u)-2.f*cos(3.f*u)-cos(4.f*u));
p.z = 0.f;
}

void thing(float R, float t, float4 a, float4& p)
{
p.x = R*(sin(t)+a.x*sin(a.y*t));
p.y = R*(cos(t)-a.x*cos(a.y*t));
p.z = R*(-sin(a.z*t));
}

void moebius(float R, float u, float v, float s, float du, float dv, float4& p )
{
p.x = 2.f*R*(cos(u)+v*cos(u/2)*cos(u));
p.y = 2.f*R*(sin(u)+v*cos(u/2)*sin(u));
p.z = 2.f*R*(v*sin(u/2));
}