Click here to Skip to main content
Click here to Skip to main content

Simple Ray Tracing in C# Part VI (Vertex Normal Interpolation)

, 9 Apr 2013
Rate this:
Please Sign up or sign in to vote.
An approach to interpolate vertex normals on triangles

Screenshot - teapota.png
fig. 1 - one normal per triangle

Screenshot - teapotb.png
fig. 2 - interpolated normals

Screenshot - mirrorteapot.png
fig. 3 - testing with reflective material

Introduction

In this new article in our ray-tracing series we will start handling more complex objects. These objects are formed by points, vertices and polygons as we have seen before.

Polygons as flat surfaces have one normal vector, and this vector against the light vectors and viewer vector gives us its color. So, working with one normal per polygon face gives a non linear result as we see in fig. 1. To solve this I have implemented an algorithm to obtain normals at each vertex and them for each ray hitpoint provide a normal interpolation between the triangle vertices giving us different normals on each polygon intersection, and hence a linear colouring between the vertices.

Note: All the articles I am posting here are for learning purposes, so do not worry at any point about speed or optimization. Instead we will keep that for more advanced articles, so please do not focus on improvements at this point.

Background

At first I recommend a previous reading in the last ray tracing articles.

Calculating normal vector of triangles

From linear algebra a normal vector of a plane can be obtained by proceeding a cross product between two vectors of this plane.

In our situation a triangle is ideal, by definition 3 points gives us a unique plane.

We have P1, P2 and P3, it is easy to get 2 vectors from this:

a = P1-P2

b = P3-P2 (remember we have 3 coordinates for each point x,y,z)

Screenshot - matrix.png
Fig 3 - cross product matrix

The cross product is given by solving the matrix given by fig. 3:

public static void Cross3(double ax, double ay, double az,
            double bx, double by, double bz,
            ref double outx, ref double outy, ref double outz)
        {
        outx = ay * bz - az * by;
        outy = az * bx - ax * bz;
        outz = ax * by - ay * bx;
        }

At this point our rendering has the fig. 1 result. Now lets think about how to calculate the interpolated normals.

At first we need to get normals for each triangle vertex, but how to achieve this?

Once we already have one normal per triangle, obtained by cross product, lets look to our points grid.

Each point belongs to more than one triangle, so it is easy to get a normal composition for each point by just adding each triangle normal on this. Aha, now we have normals at vertices!

But it is not enough... How to interpolate a hitpoint inside a triangle if we have just the vertices normals?

It is simple, we need to calculate in a similar way as we calculated the proportions for textures in our last article.

If we get u and v proportions for a given point of intersection P the interpolation is given by:

(1.0 - (u + v)) * P2 Normal + P1 Normal * u + P3 Normal * v

or better:

 nx = -( (1.0 - (u + v)) * tData.normalVector(nb).x + 
       tData.normalVector(na).x * u + tData.normalVector(nc).x * v);
 ny = -( (1.0 - (u + v)) * tData.normalVector(nb).y + 
       tData.normalVector(na).y * u + tData.normalVector(nc).y * v);
 nz = -( (1.0 - (u + v)) * tData.normalVector(nb).z + 
       tData.normalVector(na).z * u + tData.normalVector(nc).z * v); 

The function

public override void getNormal(double x, double y, double z, 
                               ref double nx, ref double ny, 
                               ref double nz)
        {
            /* code for non interpolated normals
            {
                nx = -tnormal.x;
                ny = -tnormal.y;
                nz = -tnormal.z;
                return;
            }
            */

            tPoint U = new tPoint(tData.point(va).x - tData.point(vb).x, 
                           tData.point(va).y - tData.point(vb).y, 
                           tData.point(va).z - tData.point(vb).z);
            tPoint V = new tPoint(tData.point(vc).x - tData.point(vb).x, 
                           tData.point(vc).y - tData.point(vb).y, 
                           tData.point(vc).z - tData.point(vb).z);
           
            tPoint N = new tPoint(x - tData.point(vb).x,y - 
                           tData.point(vb).y,z - tData.point(vb).z);

            double dU = tAlgebra.modv(U.x, U.y, U.z);
            double dV = tAlgebra.modv(V.x, V.y, V.z);
            double dN = tAlgebra.modv(N.x, N.y, N.z);

            tAlgebra.Normalize(ref N.x, ref N.y, ref N.z);
            tAlgebra.Normalize(ref U.x, ref U.y, ref U.z);

            double cost = tAlgebra.Dot3(N.x, N.y, N.z, U.x, U.y, U.z);
            if (cost < 0) cost = 0;
            if (cost > 1) cost = 1;

            double t = Math.Acos(cost);

            double distY = 0, distX = 0;
            distX = dN * Math.Cos(t);
            distY = dN * Math.Sin(t);

            double u = distX/ dU;
            double v = distY/ dV;

            tAlgebra.Normalize(ref tData.normalVector(na).x, 
                          ref tData.normalVector(na).y, 
                          ref tData.normalVector(na).z);
            tAlgebra.Normalize(ref tData.normalVector(nb).x, 
                          ref tData.normalVector(nb).y, 
                          ref tData.normalVector(nb).z);
            tAlgebra.Normalize(ref tData.normalVector(nc).x, 
                          ref tData.normalVector(nc).y, 
                          ref tData.normalVector(nc).z);

            nx = -( (1.0 - (u + v)) * tData.normalVector(nb).x + 
                     tData.normalVector(na).x * u + 
                     tData.normalVector(nc).x * v);
            ny = -( (1.0 - (u + v)) * tData.normalVector(nb).y + 
                     tData.normalVector(na).y * u + 
                     tData.normalVector(nc).y * v);
            nz = -( (1.0 - (u + v)) * tData.normalVector(nb).z + 
                     tData.normalVector(na).z * u + 
                     tData.normalVector(nc).z * v);
        }
 

Using the code

The samples includding classes and obj file are inside the zip file for download on the top of the article.

Screenshot - cube.png

Screenshot - cubeII.png

Screenshot - dragon.png

Screenshot - teapot_mirror2.png

License

This article, along with any associated source code and files, is licensed under The GNU General Public License (GPLv3)

Share

About the Author

andalmeida
Engineer IBM
Brazil Brazil
Senior Analyst
 
Founder of TIHunter Vagas de TI
 
Linkedin Profile
Follow on   LinkedIn

Comments and Discussions

 
SuggestionI would say it's an excellent effort, but ............ PinmentorMd. Marufuzzaman8-Apr-13 7:33 
GeneralMy vote of 5 Pinmembermanoj kumar choubey26-Feb-12 21:45 
QuestionLatest source code and library Pinmemberandalmeida6-Jan-12 5:13 

General General    News News    Suggestion Suggestion    Question Question    Bug Bug    Answer Answer    Joke Joke    Rant Rant    Admin Admin   

Use Ctrl+Left/Right to switch messages, Ctrl+Up/Down to switch threads, Ctrl+Shift+Left/Right to switch pages.

| Advertise | Privacy | Mobile
Web02 | 2.8.140827.1 | Last Updated 9 Apr 2013
Article Copyright 2007 by andalmeida
Everything else Copyright © CodeProject, 1999-2014
Terms of Service
Layout: fixed | fluid