How do I find the circumcenter of a triangle in 3D?

Question

I want to find the circumcenter of a triangle. Wolfram only shows how to find the circumcircle of a triangle in R². How can I find the circumcenter of a triangle in R³?

Answer 1

The circumcenter of a triangle can be found by the following formula, which I mined from an old posting by Jonathan Shewchuk from the Geometry Junkyard

$$\begin{align} &\text{Triangle in } \Bbb R^3\text{:}\\ &m = a + \frac {\lVert c-a\rVert^2 \left[(b-a) \times (c - a) \right] \times (b-a) + \lVert b-a\rVert^2 \left[(c-a) \times (b - a) \right] \times (c-a)} {2 \lVert (b - a) \times (c - a) \rVert^2} \end{align}$$

Where \$m\$ is the circumcenter of the triangle.

Some C++ code, given Vector3f's with overloaded +, -,

Vector3f a,b,c // are the 3 pts of the tri

Vector3f ac = c - a ;
Vector3f ab = b - a ;
Vector3f abXac = ab.cross( ac ) ;

// this is the vector from a TO the circumsphere center
Vector3f toCircumsphereCenter = (abXac.cross( ab )*ac.len2() + ac.cross( abXac )*ab.len2()) / (2.f*abXac.len2()) ;
float circumsphereRadius = toCircumsphereCenter.len() ;

// The 3 space coords of the circumsphere center then:
Vector3f ccs = a  +  toCircumsphereCenter ; // now this is the actual 3space location

Here is a picture of a triangle and its circumsphere

Answer 2

JavaScript - MY WORK - 5 TRIANGLE CENTERS IN 3D

function incenter(A,B,C) {
    var a=vlen(subv(A,B)),b=vlen(subv(B,C)),c=vlen(subv(C,A));
    var x = multv(C,a), y = multv(A,b), z = multv(B,c);
    return divv(addallv(x,y,z),a+b+c);
}

function centroid(A,B,C) {
    return divv(addallv(A,B,C),3);
}

function midpoint(A,B) {
    return divv(addv(A,B),2);
}

function orthocenter(A,B,C) {
    var c = circumcenter(A,B,C);
    return addallv(subv(A,c),subv(B,c),subv(C,c),c);
}

function circumcenter(A,B,C) {
    var z = crossprod(subv(C,B),subv(A,B));
    var a=vlen(subv(A,B)),b=vlen(subv(B,C)),c=vlen(subv(C,A));
    var r = ((b*b + c*c - a*a)/(2*b*c)) * outeradius(a,b,c);
    return addv(midpoint(A,B),multv(normaliz(crossprod(subv(A,B),z)),r));
}

function nine_point(A,B,C) {
    return midpoint(orthocenter(A,B,C),circumcenter(A,B,C));
}

function outeradius(a,b,c) { /// 3 lens
    return (a*b*c) / (4*sss_area(a,b,c));
}

function sss_area(a,b,c) {
    var sp = (a+b+c)*0.5;
    return Math.sqrt(sp*(sp-a)*(sp-b)*(sp-c));
    //semi perimeter
}

If you need the remaining functions, you can fint them in this working code example - hit the pyramid button twice.

Answer 3

//Unity/C#
Vector3 ac = c - a;
Vector3 ab = b - a;
Vector3 abXac = Vector3.Cross(ab, ac);

// this is the vector from a TO the circumsphere center
Vector3 toCircumsphereCenter = (Vector3.Cross(abXac,ab) * ac.sqrMagnitude + Vector3.Cross(ac,abXac) * ab.sqrMagnitude) / (2f * abXac.sqrMagnitude);
float circumsphereRadius = toCircumsphereCenter.magnitude;

// The 3 space coords of the circumsphere center then:
Vector3 ccs = a + toCircumsphereCenter; // now this is the actual 3space location