2D Distance Functions
Primitives
所有基本体都在原点处居中。您必须转换点以获得任意旋转、转换和缩放的对象（请参见下文）。
圆-精确
float sdCircle( vec2 p, float r )
{
return length(p) - r;
}
线 - 精确 (https://www.shadertoy.com/view/lsXGz8)
float sdLine( in vec2 p, in vec2 a, in vec2 b )
{
vec2 pa = p-a, ba = b-a;
float h = clamp( dot(pa,ba)/dot(ba,ba), 0.0, 1.0 );
return length( pa - ba*h );
}
盒子 - 精确
float sdBox( in vec2 p, in vec2 b )
{
vec2 d = abs(p)-b;
return length(max(d,vec2(0))) + min(max(d.x,d.y),0.0);
}
菱形 - 精确 (https://www.shadertoy.com/view/XdXcRB)
float sdRhombus( in vec2 p, in vec2 b )
{
vec2 q = abs(p);
float h = clamp((-2.0*ndot(q,b)+ndot(b,b))/dot(b,b),-1.0,1.0);
float d = length( q - 0.5*b*vec2(1.0-h,1.0+h) );
return d * sign( q.x*b.y + q.y*b.x - b.x*b.y );
}
等边三角形 - 精确 (https://www.shadertoy.com/view/Xl2yDW)
float sdEquilateralTriangle( in vec2 p )
{
const float k = sqrt(3.0);
p.x = abs(p.x) - 1.0;
p.y = p.y + 1.0/k;
if( p.x+k*p.y>0.0 ) p = vec2(p.x-k*p.y,-k*p.x-p.y)/2.0;
p.x -= clamp( p.x, -2.0, 0.0 );
return -length(p)*sign(p.y);
}
等腰三角形 - exact (https://www.shadertoy.com/view/MldcD7)
float sdTriangleIsosceles( in vec2 p, in vec2 q )
{
p.x = abs(p.x);
vec2 a = p - q*clamp( dot(p,q)/dot(q,q), 0.0, 1.0 );
vec2 b = p - q*vec2( clamp( p.x/q.x, 0.0, 1.0 ), 1.0 );
float s = -sign( q.y );
vec2 d = min( vec2( dot(a,a), s*(p.x*q.y-p.y*q.x) ), vec2( dot(b,b), s*(p.y-q.y) ));
return -sqrt(d.x)*sign(d.y);
}不均匀的胶囊 - exact (https://www.shadertoy.com/view/4lcBWn)
float sdUnevenCapsule( vec2 p, float r1, float r2, float h )
{
p.x = abs(p.x);
float b = (r1-r2)/h;
float a = sqrt(1.0-b*b);
float k = dot(p,vec2(-b,a));
if( k < 0.0 ) return length(p) - r1;
if( k > a*h ) return length(p-vec2(0.0,h)) - r2; return dot(p, vec2(a,b) ) - r1;
}
三角形 - exact (https://www.shadertoy.com/view/XsXSz4)
float sdTriangle( in vec2 p, in vec2 p0, in vec2 p1, in vec2 p2 )
{
vec2 e0 = p1-p0, e1 = p2-p1, e2 = p0-p2;
vec2 v0 = p -p0, v1 = p -p1, v2 = p -p2;
vec2 pq0 = v0 - e0*clamp( dot(v0,e0)/dot(e0,e0), 0.0, 1.0 );
vec2 pq1 = v1 - e1*clamp( dot(v1,e1)/dot(e1,e1), 0.0, 1.0 );
vec2 pq2 = v2 - e2*clamp( dot(v2,e2)/dot(e2,e2), 0.0, 1.0 );
float s = sign( e0.x*e2.y - e0.y*e2.x );
vec2 d = min(min(vec2(dot(pq0,pq0), s*(v0.x*e0.y-v0.y*e0.x)), vec2(dot(pq1,pq1), s*(v1.x*e1.y-v1.y*e1.x))), vec2(dot(pq2,pq2), s*(v2.x*e2.y-v2.y*e2.x)));
return -sqrt(d.x)*sign(d.y);
}
正五边形 - exact (https://www.shadertoy.com/view/llVyWW)
float sdPentagon( in vec2 p, in float r )
{
const vec3 k = vec3(0.809016994,0.587785252,0.726542528);
p.x = abs(p.x);
p -= 2.0*min(dot(vec2(-k.x,k.y),p),0.0)*vec2(-k.x,k.y);
p -= 2.0*min(dot(vec2( k.x,k.y),p),0.0)*vec2( k.x,k.y);
p -= vec2(clamp(p.x,-r*k.z,r*k.z),r);
return length(p)*sign(p.y);
}
正六边形 - exact
float sdHexagon( in vec2 p, in float r )
{
const vec3 k = vec3(-0.866025404,0.5,0.577350269);
p = abs(p);
p -= 2.0*min(dot(k.xy,p),0.0)*k.xy;
p -= vec2(clamp(p.x, -k.z*r, k.z*r), r);
return length(p)*sign(p.y);
}
八边形 - exact (https://www.shadertoy.com/view/llGfDG)
float sdOctogon( in vec2 p, in float r )
{
const vec3 k = vec3(-0.9238795325, 0.3826834323, 0.4142135623 );
p = abs(p);
p -= 2.0*min(dot(vec2( k.x,k.y),p),0.0)*vec2( k.x,k.y);
p -= 2.0*min(dot(vec2(-k.x,k.y),p),0.0)*vec2(-k.x,k.y);
p -= vec2(clamp(p.x, -k.z*r, k.z*r), r);
return length(p)*sign(p.y);
}
六角星形 - exact (https://www.shadertoy.com/view/tt23RR)
float sdHexagram( in vec2 p, in float r )
{
const vec4 k = vec4(-0.5,0.8660254038,0.5773502692,1.7320508076);
p = abs(p);
p -= 2.0*min(dot(k.xy,p),0.0)*k.xy;
p -= 2.0*min(dot(k.yx,p),0.0)*k.yx;
p -= vec2(clamp(p.x,r*k.z,r*k.w),r);
return length(p)*sign(p.y);
}五角星形 - exact (https://www.shadertoy.com/view/3tSGDy)
float sdStar(in vec2 p, in float r, in int n, in float m)
{
// next 4 lines can be precomputed for a given shape
float an = 3.141593/float(n);
float en = 6.283185/m;
vec2 acs = vec2(cos(an),sin(an));
vec2 ecs = vec2(cos(en),sin(en));
// ecs=vec2(0,1) for regular polygon,
float bn = mod(atan(p.x,p.y),2.0*an) - an;
p = length(p)*vec2(cos(bn),abs(sin(bn)));
p -= r*acs;
p += ecs*clamp( -dot(p,ecs), 0.0, r*acs.y/ecs.y);
return length(p)*sign(p.x);
}
等腰梯形 - exact (https://www.shadertoy.com/view/MlycD3)
float sdTrapezoid( in vec2 p, in float r1, float r2, float he )
{
vec2 k1 = vec2(r2,he); vec2 k2 = vec2(r2-r1,2.0*he);
p.x = abs(p.x);
vec2 ca = vec2(p.x-min(p.x,(p.y<0.0)?r1:r2), abs(p.y)-he);
vec2 cb = p - k1 + k2*clamp( dot(k1-p,k2)/dot2(k2), 0.0, 1.0 );
float s = (cb.x<0.0 && ca.y<0.0) ? -1.0 : 1.0;
return s*sqrt( min(dot2(ca),dot2(cb)) );
}
饼状图 - exact (https://www.shadertoy.com/view/3l23RK)
float sdPie( in vec2 p, in vec2 c, in float r )
{
p.x = abs(p.x);
float l = length(p) - r;
float m = length(p-c*clamp(dot(p,c),0.0,r));
// c = sin/cos of the aperture
return max(l,m*sign(c.y*p.x-c.x*p.y));
}
弧形 - exact (https://www.shadertoy.com/view/wl23RK)
float sdArc( in vec2 p, in vec2 sca, in vec2 scb, in float ra, float rb )
{
p *= mat2(sca.x,sca.y,-sca.y,sca.x);
p.x = abs(p.x);
float k = (scb.y*p.x>scb.x*p.y) ? dot(p.xy,scb) : length(p.xy);
return sqrt( dot(p,p) + ra*ra - 2.0*ra*k ) - rb;
}
马蹄形 - exact (https://www.shadertoy.com/view/WlSGW1)
float sdHorseshoe( in vec2 p, in vec2 c, in float r, in vec2 w )
{
p.x = abs(p.x);
float l = length(p);
p = mat2(-c.x, c.y, c.y, c.x)*p;
p = vec2((p.y>0.0)?p.x:l*sign(-c.x), (p.x>0.0)?p.y:l );
p = vec2(p.x,abs(p.y-r))-w;
return length(max(p,0.0)) + min(0.0,max(p.x,p.y));
}
胶囊形 - exact (https://www.shadertoy.com/view/XtVfRW)
float sdVesica(vec2 p, float r, float d)
{
p = abs(p);
float b = sqrt(r*r-d*d);
return ((p.y-b)*d>p.x*b) ? length(p-vec2(0.0,b)) : length(p-vec2(-d,0.0))-r;
}
十字形 - exact (https://www.shadertoy.com/view/XtGfzw)
float sdCross( in vec2 p, in vec2 b, float r )
{
p = abs(p);
p = (p.y>p.x) ? p.yx : p.xy;
vec2 q = p - b;
float k = max(q.y,q.x); vec2 w = (k>0.0) ? q : vec2(b.y-p.x,-k);
return sign(k)*length(max(w,0.0)) + r;
}
多边形 - exact (https://www.shadertoy.com/view/wdBXRW)
float sdPolygon( in vec2[N] v, in vec2 p )
{
float d = dot(p-v[0],p-v[0]);
float s = 1.0;
for( int i=0, j=N; i=v[i].y,p.ye.y*w.x);
if( all(c) || all(not(c)) ) s*=-1.0; }
return s*sqrt(d);
}
椭圆 - exact (https://www.shadertoy.com/view/4sS3zz)
float sdEllipse( in vec2 p, in vec2 ab )
{
p = abs(p);
if( p.x > p.y )
{
p=p.yx;ab=ab.yx;
}
float l = ab.y*ab.y - ab.x*ab.x;
float m = ab.x*p.x/l;
float m2 = m*m;
float n = ab.y*p.y/l;
float n2 = n*n;
float c = (m2+n2-1.0)/3.0;
float c3 = c*c*c;
float q = c3 + m2*n2*2.0;
float d = c3 + m2*n2;
float g = m + m*n2;
float co;
if( d<0.0 )
{
float h = acos(q/c3)/3.0;
float s = cos(h);
float t = sin(h)*sqrt(3.0);
float rx = sqrt( -c*(s + t + 2.0) + m2 );
float ry = sqrt( -c*(s - t + 2.0) + m2 );
co = (ry+sign(l)*rx+abs(g)/(rx*ry)- m)/2.0;
} else {
float h = 2.0*m*n*sqrt( d );
float s = sign(q+h)*pow(abs(q+h), 1.0/3.0);
float u = sign(q-h)*pow(abs(q-h), 1.0/3.0);
float rx = -s - u - c*4.0 + 2.0*m2;
float ry = (s - u)*sqrt(3.0);
float rm = sqrt( rx*rx + ry*ry );
co = (ry/sqrt(rm-rx)+2.0*g/rm-m)/2.0;
}
vec2 r = ab * vec2(co, sqrt(1.0-co*co));
return length(r-p) * sign(p.y-r.y);
}
抛物线 - exact (https://www.shadertoy.com/view/ws3GD7)
float sdParabola( in vec2 pos, in float k )
{
pos.x = abs(pos.x);
float p = (1.0-2.0*k*pos.y)/(6.0*k*k);
float q = -abs(pos.x)/(4.0*k*k);
float h = q*q + p*p*p;
float r = sqrt(abs(h));
float x = (h>0.0) ? pow(-q+r,1.0/3.0) - pow(abs(-q-r),1.0/3.0)*sign(q+r) : 2.0*cos(atan(r,-q)/3.0)*sqrt(-p);
return length(pos-vec2(x,k*x*x)) * sign(pos.x-x);
}
二次贝塞尔曲线 - exact (https://www.shadertoy.com/view/MlKcDD)
float sdBezier( in vec2 pos, in vec2 A, in vec2 B, in vec2 C )
{
vec2 a = B - A;
vec2 b = A - 2.0*B + C;
vec2 c = a * 2.0;
vec2 d = A - pos;
float kk = 1.0 / dot(b,b);
float kx = kk * dot(a,b);
float ky = kk * (2.0*dot(a,a)+dot(d,b)) / 3.0;
float kz = kk * dot(d,a);
float res = 0.0;
float p = ky - kx*kx;
float p3 = p*p*p;
float q = kx*(2.0*kx*kx - 3.0*ky) + kz;
float h = q*q + 4.0*p3;
if( h >= 0.0) {
h = sqrt(h);
vec2 x = (vec2(h, -h) - q) / 2.0;
vec2 uv = sign(x)*pow(abs(x), vec2(1.0/3.0));
float t = uv.x + uv.y - kx;
t = clamp( t, 0.0, 1.0 );
vec2 qos = d + (c + b*t)*t;
res = dot(qos,qos);
} else {
float z = sqrt(-p);
float v = acos( q/(p*z*2.0) ) / 3.0;
float m = cos(v);
float n = sin(v)*1.732050808;
vec3 t = vec3(m + m, -n - m, n - m) * z - kx;
t = clamp( t, 0.0, 1.0 );
vec2 qos = d + (c + b*t.x)*t.x;
res = dot(qos,qos);
qos = d + (c + b*t.y)*t.y;
res = min(res,dot(qos,qos));
qos = d + (c + b*t.z)*t.z;
res = min(res,dot(qos,qos));
}
return sqrt( res );
}Making shapes rounded
使形状变圆
All the shapes above can be converted into rounded shapes by subtracting a constant from their distance function. That, effectivelly moves the isosurface (isopetimeter I guess) from the level zero to one of the outter rings, which naturally are rounded, as it can be seen in the yellow areas in all the images above. So, basically, for any shape defined by d(x,y) = sdf(x,y), one can make it sounded by computing d(x,y) = sdf(x,y) - r:
float sdRoundedShape( in vec2 p, in float r )
{
return sdShape(p) - r;
}These are a few examples: rounded line, rounded triangle, rounded box and a rounded pentagon:
Making shapes annular
使形状变环状的
Similarly, shapes can be made annular (like a ring), but taking their absolute value and then substracting a constant from their field. So, for any shape defined by d(x,y) = sdf(x,y) compute d(x,y) = |sdf(x,y)| - r:
float sdAnnularShape( in vec2 p, in float r )
{
return abs(sdShape(p)) - r;
}These are a few examples: annular rounded line, an annular triangle, an annular box and a annular pentagon:
Rigid deformations, domain repetitiion, range distortion, boolean operations, smooth connections
All of these work just as well as in 3D case, so I won't repeat the examples already exist in this article.