Constructing a unit normal vector to curve
What we're building to
- A unit normal vector to a two-dimensional curve is a vector with magnitude that is perpendicular to the curve at some point.
- Typically you look for a function that gives you all possible unit normal vectors of a given curve, not just one vector.
- To find the unit normal vector of a two-dimensional curve, take the following steps:
- Find the tangent vector, which requires taking the derivative of the parametric function defining the curve.
- Rotate that tangent vector , which involves swapping the coordinates and making one of them negative.
- Normalize the result, which requires dividing it by its own magnitude.
- Abstractly speaking, the result you get will look something like this:For a given tiny step along the curve, think of as the -component of that step, as the -component of that step, and as the length of that step.
Example: Normal vectors to a sine curve
Step 0: Parameterize
Step 1: Find a tangent vector
Step 2: Rotate this vector
Step 3: Scale it to magnitude
- Step 0: Make sure the curve is given parametrically
- Step 1: Find a tangent vector to your curve by differentiating the parametric function:
- Step 2: Rotate this vector by swapping the coordinates and making one negative.
- Step 3: To make this a unit normal vector, divide it by its magnitude: