How To Draw A Frustum in OpenGL

Question

I created a frustum class using C# and OpenTK, and it works perfectly. However, I want to be able to draw the bounds of the frustum for debugging purposes. I have no idea how to get the vertex data from the Matrix data for drawing though. I was just wondering if anyone did know how to do it.

Here is my Frustum class:

using System;
using GrimoireGDK.GameDevelopmentKit.Math.Geometry;
using OpenTK;

namespace GrimoireGameClient.Client.Window.Bindings {
    public class Frustum {
        private float[] _clipMatrix = new float[ 16 ];
        private float[ , ] _frustum = new float[ 6 , 4 ];

        public const int A = 0;
        public const int B = 1;
        public const int C = 2;
        public const int D = 3;

        public enum ClippingPlane : int {
            Right = 0 ,
            Left = 1 ,
            Bottom = 2 ,
            Top = 3 ,
            Back = 4 ,
            Front = 5
        }

        private void NormalizePlane( float[ , ] frustum , int side ) {
            float magnitude = ( float )Math.Sqrt( ( frustum[ side , 0 ] * frustum[ side , 0 ] ) + ( frustum[ side , 1 ] * frustum[ side , 1 ] )
                                                + ( frustum[ side , 2 ] * frustum[ side , 2 ] ) );
            frustum[ side , 0 ] /= magnitude;
            frustum[ side , 1 ] /= magnitude;
            frustum[ side , 2 ] /= magnitude;
            frustum[ side , 3 ] /= magnitude;
        }

        public bool PointVsFrustum( float x , float y , float z ) {
            for( int i = 0 ; i < 6 ; i++ ) {
                if( this._frustum[ i , 0 ] * x + this._frustum[ i , 1 ] * y + this._frustum[ i , 2 ] * z + this._frustum[ i , 3 ] <= 0.0f ) {
                    return false;
                }
            }
            return true;
        }

        public bool PointVsFrustum( Vector3 location ) {
            for( int i = 0 ; i < 6 ; i++ ) {
                if( this._frustum[ i , 0 ] * location.X + this._frustum[ i , 1 ] * location.Y + this._frustum[ i , 2 ] * location.Z + this._frustum[ i , 3 ] <= 0.0f ) {
                    return false;
                }
            }
            return true;
        }

        public bool SphereVsFrustum( float x , float y , float z , float radius ) {
            float d = 0;
            for( int p = 0 ; p < 6 ; p++ ) {
                d = _frustum[ p , 0 ] * x + _frustum[ p , 1 ] * y + _frustum[ p , 2 ] * z + _frustum[ p , 3 ];
                if( d <= -radius ) {
                    return false;
                }
            }
            return true;
        }

        public bool SphereVsFrustum( Vector3 location , float radius ) {
            float d = 0;
            for( int p = 0 ; p < 6 ; p++ ) {
                d = _frustum[ p , 0 ] * location.X + _frustum[ p , 1 ] * location.Y + _frustum[ p , 2 ] * location.Z + _frustum[ p , 3 ];
                if( d <= -radius ) {
                    return false;
                }
            }
            return true;
        }

        public bool VolumeVsFrustum( float x , float y , float z , float width , float height , float length ) {
            for( int i = 0 ; i < 6 ; i++ ) {
                if( _frustum[ i , A ] * ( x - width ) + _frustum[ i , B ] * ( y - height ) + _frustum[ i , C ] * ( z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + width ) + _frustum[ i , B ] * ( y - height ) + _frustum[ i , C ] * ( z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x - width ) + _frustum[ i , B ] * ( y + height ) + _frustum[ i , C ] * ( z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + width ) + _frustum[ i , B ] * ( y + height ) + _frustum[ i , C ] * ( z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x - width ) + _frustum[ i , B ] * ( y - height ) + _frustum[ i , C ] * ( z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + width ) + _frustum[ i , B ] * ( y - height ) + _frustum[ i , C ] * ( z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x - width ) + _frustum[ i , B ] * ( y + height ) + _frustum[ i , C ] * ( z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + width ) + _frustum[ i , B ] * ( y + height ) + _frustum[ i , C ] * ( z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                return false;
            }
            return true;
        }

        public bool VolumeVsFrustum( Vector3 location, float width , float height , float length ) {
            for( int i = 0 ; i < 6 ; i++ ) {
                if( _frustum[ i , A ] * ( location.X - width ) + _frustum[ i , B ] * ( location.Y - height ) + _frustum[ i , C ] * ( location.Z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + width ) + _frustum[ i , B ] * ( location.Y - height ) + _frustum[ i , C ] * ( location.Z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X - width ) + _frustum[ i , B ] * ( location.Y + height ) + _frustum[ i , C ] * ( location.Z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + width ) + _frustum[ i , B ] * ( location.Y + height ) + _frustum[ i , C ] * ( location.Z - length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X - width ) + _frustum[ i , B ] * ( location.Y - height ) + _frustum[ i , C ] * ( location.Z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + width ) + _frustum[ i , B ] * ( location.Y - height ) + _frustum[ i , C ] * ( location.Z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X - width ) + _frustum[ i , B ] * ( location.Y + height ) + _frustum[ i , C ] * ( location.Z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + width ) + _frustum[ i , B ] * ( location.Y + height ) + _frustum[ i , C ] * ( location.Z + length ) + _frustum[ i , D ] > 0 )
                    continue;
                return false;
            }
            return true;
        }

        public bool VolumeVsFrustum( Vector3 location , BoundingVolume volume ) {
            for( int i = 0 ; i < 6 ; i++ ) {
                if( _frustum[ i , A ] * ( location.X - volume.Width ) + _frustum[ i , B ] * ( location.Y - volume.Height ) + _frustum[ i , C ] * ( location.Z - volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + volume.Width ) + _frustum[ i , B ] * ( location.Y - volume.Height ) + _frustum[ i , C ] * ( location.Z - volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X - volume.Width ) + _frustum[ i , B ] * ( location.Y + volume.Height ) + _frustum[ i , C ] * ( location.Z - volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + volume.Width ) + _frustum[ i , B ] * ( location.Y + volume.Height ) + _frustum[ i , C ] * ( location.Z - volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X - volume.Width ) + _frustum[ i , B ] * ( location.Y - volume.Height ) + _frustum[ i , C ] * ( location.Z + volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + volume.Width ) + _frustum[ i , B ] * ( location.Y - volume.Height ) + _frustum[ i , C ] * ( location.Z + volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X - volume.Width ) + _frustum[ i , B ] * ( location.Y + volume.Height ) + _frustum[ i , C ] * ( location.Z + volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( location.X + volume.Width ) + _frustum[ i , B ] * ( location.Y + volume.Height ) + _frustum[ i , C ] * ( location.Z + volume.Length ) + _frustum[ i , D ] > 0 )
                    continue;
                return false;
            }
            return true;
        }

        public bool CubeVsFrustum( float x , float y , float z , float size ) {
            for( int i = 0 ; i < 6 ; i++ ) {
                if( _frustum[ i , A ] * ( x - size ) + _frustum[ i , B ] * ( y - size ) + _frustum[ i , C ] * ( z - size ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + size ) + _frustum[ i , B ] * ( y - size ) + _frustum[ i , C ] * ( z - size ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x - size ) + _frustum[ i , B ] * ( y + size ) + _frustum[ i , C ] * ( z - size ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + size ) + _frustum[ i , B ] * ( y + size ) + _frustum[ i , C ] * ( z - size ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x - size ) + _frustum[ i , B ] * ( y - size ) + _frustum[ i , C ] * ( z + size ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + size ) + _frustum[ i , B ] * ( y - size ) + _frustum[ i , C ] * ( z + size ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x - size ) + _frustum[ i , B ] * ( y + size ) + _frustum[ i , C ] * ( z + size ) + _frustum[ i , D ] > 0 )
                    continue;
                if( _frustum[ i , A ] * ( x + size ) + _frustum[ i , B ] * ( y + size ) + _frustum[ i , C ] * ( z + size ) + _frustum[ i , D ] > 0 )
                    continue;
                return false;
            }
            return true;
        }


        public void CalculateFrustum( Matrix4 projectionMatrix , Matrix4 modelViewMatrix ) {
            _clipMatrix[ 0 ] = ( modelViewMatrix.M11 * projectionMatrix.M11 ) + ( modelViewMatrix.M12 * projectionMatrix.M21 ) + ( modelViewMatrix.M13 * projectionMatrix.M31 ) + ( modelViewMatrix.M14 * projectionMatrix.M41 );
            _clipMatrix[ 1 ] = ( modelViewMatrix.M11 * projectionMatrix.M12 ) + ( modelViewMatrix.M12 * projectionMatrix.M22 ) + ( modelViewMatrix.M13 * projectionMatrix.M32 ) + ( modelViewMatrix.M14 * projectionMatrix.M42 );
            _clipMatrix[ 2 ] = ( modelViewMatrix.M11 * projectionMatrix.M13 ) + ( modelViewMatrix.M12 * projectionMatrix.M23 ) + ( modelViewMatrix.M13 * projectionMatrix.M33 ) + ( modelViewMatrix.M14 * projectionMatrix.M43 );
            _clipMatrix[ 3 ] = ( modelViewMatrix.M11 * projectionMatrix.M14 ) + ( modelViewMatrix.M12 * projectionMatrix.M24 ) + ( modelViewMatrix.M13 * projectionMatrix.M34 ) + ( modelViewMatrix.M14 * projectionMatrix.M44 );

            _clipMatrix[ 4 ] = ( modelViewMatrix.M21 * projectionMatrix.M11 ) + ( modelViewMatrix.M22 * projectionMatrix.M21 ) + ( modelViewMatrix.M23 * projectionMatrix.M31 ) + ( modelViewMatrix.M24 * projectionMatrix.M41 );
            _clipMatrix[ 5 ] = ( modelViewMatrix.M21 * projectionMatrix.M12 ) + ( modelViewMatrix.M22 * projectionMatrix.M22 ) + ( modelViewMatrix.M23 * projectionMatrix.M32 ) + ( modelViewMatrix.M24 * projectionMatrix.M42 );
            _clipMatrix[ 6 ] = ( modelViewMatrix.M21 * projectionMatrix.M13 ) + ( modelViewMatrix.M22 * projectionMatrix.M23 ) + ( modelViewMatrix.M23 * projectionMatrix.M33 ) + ( modelViewMatrix.M24 * projectionMatrix.M43 );
            _clipMatrix[ 7 ] = ( modelViewMatrix.M21 * projectionMatrix.M14 ) + ( modelViewMatrix.M22 * projectionMatrix.M24 ) + ( modelViewMatrix.M23 * projectionMatrix.M34 ) + ( modelViewMatrix.M24 * projectionMatrix.M44 );

            _clipMatrix[ 8 ] = ( modelViewMatrix.M31 * projectionMatrix.M11 ) + ( modelViewMatrix.M32 * projectionMatrix.M21 ) + ( modelViewMatrix.M33 * projectionMatrix.M31 ) + ( modelViewMatrix.M34 * projectionMatrix.M41 );
            _clipMatrix[ 9 ] = ( modelViewMatrix.M31 * projectionMatrix.M12 ) + ( modelViewMatrix.M32 * projectionMatrix.M22 ) + ( modelViewMatrix.M33 * projectionMatrix.M32 ) + ( modelViewMatrix.M34 * projectionMatrix.M42 );
            _clipMatrix[ 10 ] = ( modelViewMatrix.M31 * projectionMatrix.M13 ) + ( modelViewMatrix.M32 * projectionMatrix.M23 ) + ( modelViewMatrix.M33 * projectionMatrix.M33 ) + ( modelViewMatrix.M34 * projectionMatrix.M43 );
            _clipMatrix[ 11 ] = ( modelViewMatrix.M31 * projectionMatrix.M14 ) + ( modelViewMatrix.M32 * projectionMatrix.M24 ) + ( modelViewMatrix.M33 * projectionMatrix.M34 ) + ( modelViewMatrix.M34 * projectionMatrix.M44 );

            _clipMatrix[ 12 ] = ( modelViewMatrix.M41 * projectionMatrix.M11 ) + ( modelViewMatrix.M42 * projectionMatrix.M21 ) + ( modelViewMatrix.M43 * projectionMatrix.M31 ) + ( modelViewMatrix.M44 * projectionMatrix.M41 );
            _clipMatrix[ 13 ] = ( modelViewMatrix.M41 * projectionMatrix.M12 ) + ( modelViewMatrix.M42 * projectionMatrix.M22 ) + ( modelViewMatrix.M43 * projectionMatrix.M32 ) + ( modelViewMatrix.M44 * projectionMatrix.M42 );
            _clipMatrix[ 14 ] = ( modelViewMatrix.M41 * projectionMatrix.M13 ) + ( modelViewMatrix.M42 * projectionMatrix.M23 ) + ( modelViewMatrix.M43 * projectionMatrix.M33 ) + ( modelViewMatrix.M44 * projectionMatrix.M43 );
            _clipMatrix[ 15 ] = ( modelViewMatrix.M41 * projectionMatrix.M14 ) + ( modelViewMatrix.M42 * projectionMatrix.M24 ) + ( modelViewMatrix.M43 * projectionMatrix.M34 ) + ( modelViewMatrix.M44 * projectionMatrix.M44 );

            _frustum[ ( int )ClippingPlane.Right , 0 ] = _clipMatrix[ 3 ] - _clipMatrix[ 0 ];
            _frustum[ ( int )ClippingPlane.Right , 1 ] = _clipMatrix[ 7 ] - _clipMatrix[ 4 ];
            _frustum[ ( int )ClippingPlane.Right , 2 ] = _clipMatrix[ 11 ] - _clipMatrix[ 8 ];
            _frustum[ ( int )ClippingPlane.Right , 3 ] = _clipMatrix[ 15 ] - _clipMatrix[ 12 ];
            NormalizePlane( _frustum , ( int )ClippingPlane.Right );

            _frustum[ ( int )ClippingPlane.Left , 0 ] = _clipMatrix[ 3 ] + _clipMatrix[ 0 ];
            _frustum[ ( int )ClippingPlane.Left , 1 ] = _clipMatrix[ 7 ] + _clipMatrix[ 4 ];
            _frustum[ ( int )ClippingPlane.Left , 2 ] = _clipMatrix[ 11 ] + _clipMatrix[ 8 ];
            _frustum[ ( int )ClippingPlane.Left , 3 ] = _clipMatrix[ 15 ] + _clipMatrix[ 12 ];
            NormalizePlane( _frustum , ( int )ClippingPlane.Left );

            _frustum[ ( int )ClippingPlane.Bottom , 0 ] = _clipMatrix[ 3 ] + _clipMatrix[ 1 ];
            _frustum[ ( int )ClippingPlane.Bottom , 1 ] = _clipMatrix[ 7 ] + _clipMatrix[ 5 ];
            _frustum[ ( int )ClippingPlane.Bottom , 2 ] = _clipMatrix[ 11 ] + _clipMatrix[ 9 ];
            _frustum[ ( int )ClippingPlane.Bottom , 3 ] = _clipMatrix[ 15 ] + _clipMatrix[ 13 ];
            NormalizePlane( _frustum , ( int )ClippingPlane.Bottom );

            _frustum[ ( int )ClippingPlane.Top , 0 ] = _clipMatrix[ 3 ] - _clipMatrix[ 1 ];
            _frustum[ ( int )ClippingPlane.Top , 1 ] = _clipMatrix[ 7 ] - _clipMatrix[ 5 ];
            _frustum[ ( int )ClippingPlane.Top , 2 ] = _clipMatrix[ 11 ] - _clipMatrix[ 9 ];
            _frustum[ ( int )ClippingPlane.Top , 3 ] = _clipMatrix[ 15 ] - _clipMatrix[ 13 ];
            NormalizePlane( _frustum , ( int )ClippingPlane.Top );

            _frustum[ ( int )ClippingPlane.Back , 0 ] = _clipMatrix[ 3 ] - _clipMatrix[ 2 ];
            _frustum[ ( int )ClippingPlane.Back , 1 ] = _clipMatrix[ 7 ] - _clipMatrix[ 6 ];
            _frustum[ ( int )ClippingPlane.Back , 2 ] = _clipMatrix[ 11 ] - _clipMatrix[ 10 ];
            _frustum[ ( int )ClippingPlane.Back , 3 ] = _clipMatrix[ 15 ] - _clipMatrix[ 14 ];
            NormalizePlane( _frustum , ( int )ClippingPlane.Back );

            _frustum[ ( int )ClippingPlane.Front , 0 ] = _clipMatrix[ 3 ] + _clipMatrix[ 2 ];
            _frustum[ ( int )ClippingPlane.Front , 1 ] = _clipMatrix[ 7 ] + _clipMatrix[ 6 ];
            _frustum[ ( int )ClippingPlane.Front , 2 ] = _clipMatrix[ 11 ] + _clipMatrix[ 10 ];
            _frustum[ ( int )ClippingPlane.Front , 3 ] = _clipMatrix[ 15 ] + _clipMatrix[ 14 ];
            NormalizePlane( _frustum , ( int )ClippingPlane.Front );
        }

        /// <summary>
        /// Draw the frustum for debugging purposes.
        /// </summary>
        public void Draw() {
            // Implement drawing code here.
        }
    }
}

Does anyone know how to draw the Frustum? And I don't care if it is done in immediate mode; in fact, I would almost prefer that. Here is the kind of visual output I am looking for:

Answer 1

Looks like you have the six planes making up the frustum (top, bottom, left, right, front, back) described as four components, the normal vector of the plane and the distance along that vector from the origin (xyzD).

One easy method: given any three planes you can find the relevant corner point (so top, left, back for the top-left-back corner; etc.) with a series of dot and cross products. A good reference is the MonoGame.BoundingFrustum.IntersectionPoint method (MIT License), which (before recent optimisations) goes:

Vector3 IntersectionPoint(ref Plane a, ref Plane b, ref Plane c)
{
    // Formula used
    //                d1 ( N2 * N3 ) + d2 ( N3 * N1 ) + d3 ( N1 * N2 )
    //P =   ---------------------------------------------------------------------
    //                             N1 . ( N2 * N3 )
    //
    // Note: N refers to the normal, d refers to the displacement. '.' means dot product. '*' means cross product

    Vector3 v1, v2, v3;
    float f = -Vector3.Dot(a.Normal, Vector3.Cross(b.Normal, c.Normal));

    v1 = (a.D * (Vector3.Cross(b.Normal, c.Normal)));
    v2 = (b.D * (Vector3.Cross(c.Normal, a.Normal)));
    v3 = (c.D * (Vector3.Cross(a.Normal, b.Normal)));

    Vector3 vec = new Vector3(v1.X + v2.X + v3.X, v1.Y + v2.Y + v3.Y, v1.Z + v2.Z + v3.Z);
    return vec / f;
}

You can repeat this for all eight corners, and then connect the corners together to draw the frustum as a series lines/edges.

The above link is a fairly good reference - in fact the MonoGame code, and its XNA ancestor, are both good references for geometry handling that is "fast enough", and MonoGame's MIT license is very permissive. Their only drawbacks tend to be assumptions in coordinate-system handedness, matrix multiplication order (row vs column matrices) and the like.

I'm sure it goes without saying, to get a full picture you'll want to view the results with a different camera, as in the sample visual output you give.

Answer 2

Neither of these suggestions are exactly what you are asking, but they might help you think about it.

As near as I can tell (assuming the code works, as you say it does), the _frustum arrays define the 6 planes on the sides of the frustum.

So, you could simply compute the 8 points at the corner of the volume by computing the intersection of the appropriate planes (as discussed here: Point of intersection of three planes 3D - C#)

I suspect that when you work out the equations, they will simplify down to some relatively simple expressions based on the data in clipmatrix.

Alternatively, if you know the frustum defined by your perspectivematrix (in the simple canonical coordinates relative to the viewpoint, centered along the z-axis), you can simply invert the modelViewMatrix and multiply these points by it, giving you the points back in world coordinates.