tangent, normal, and binormal vectors of curve

Given a parameterized curve , the velocity and acceleration are often expressed as a combination of two orthogonal unit vectors for each time .

We have

• as the unit tangent vector which is tangent to at time ,
• and as the unit normal vector to which is orthogonal at time , By definition, the dot product .

More precisely,

• is defined as
• is defined as Fact:

Where is the derivative of the speed, and is a constant coefficient that we called curvature (which measures how much a curve “bends”)

Binormal Vector

In 3d, the binormal vector is defined as the cross product of tangent and normal vector

Reference

• Calculus Blue Chapter 8