Determining the Normal Vector

Now we would like to understand the fundamental differential equation that governs this process. We will do this by constructing this equation.

The first step in this process is determining an equation for the normal vector . An important realization is that the vector marked in red in Figure 3 may be represented as the quantity .

We assume that the normal vector may be represented as some combination of the two vectors on either side of it. This assumption may be symbolically written

Multiplying each side of Equation 1 by , we observe

Multiplying each side of Equation 1 by , we observe

Since (because they are both unit vectors), . Substituting from the above equations, or

cannot be true because that would imply , which indicates that and intersect at an angle of 180°. Therefore, . Because the equation holds for all values of , we may arbitrarily choose the constants. We will choose . Substituting into equation 1,

Knowing the normal will help us to solve for the vector tangent to .