Calculating the reflected ray/vector

In computer graphics applications its often needed to calculate the reflection ray for example if you are writing a ray tracer, a shader for some advanced lighting , or environment mapping etc.

If you are writing shaders , there are standard library function to do that . In cg shading language there is a function reflect(also its more efficient than writing our own).

In this post rather than just giving the vector formula for reflection ray , i am trying to explain the simple mathematics behind that.

See the following image, I is the original ray, and R is the reflected ray which we need to found.N is the normal of the incident plane. P is the line perpendicular from normal to both rays. it is obvious that at both ends the length of P will be same.

Dot product between two unit vectors gives the cosine of the angle between them. So using this idea we can find

R = DotProduct[ I, N ] * N + P . ---> Eq(1)

We don't know P now. But I + P = N * DotProduct[ I,N].

So by rearranging P = N * DotProduct[ I,N] - I. Substituting the value of P now in equation(1) gives the final equation.

Here it is the final equation R = 2 * N * ( DotProduct[ I,N] ) - I

Color Image Segmentation using Meanshift Algorithm

The meanshift method can be used to segment color image. In this method image pixels is treated as points in color space.In each iteration the meanshift vector is calculated for points which are inside the kernel radius.After that the the old kernel location is changed to meanshift vector's position. Color is also updated. This process continues untill both converge.

Original Image

Meanshift Filterd Image

As the iteration count increases , the same color segment which has same type of colors will get merged together. You can see the effect of meanshift filter on sachin's photo.Its like water painting (not exactly,there are other filters for that. )

How to check a Point inside a Triangle

When i started to learn vector mathematics , i had found many methods to determine whether a point is inside a triangle or not.Easy method is not the fastest. This type of calculations are very important for making a fast graphics library.
Following are the methods which i remember now.
1. Using Cross Products

Take cross product between each triangle edge and the point to check. If the direction of cross product result is for the three edges that point is inside the triangle , otherwise not

2. Using the angle between the vectors made by point and triangle coordinates.If the sum of angle is 360 the point is inside , otherwise not.

3. Using Point and Plane test.See the figure. You need to know plane equation to understand it.

4. Using Bary centric Coordinates

This is the fastest of above. It involves checking the point in Bary centric coordinates system.