Postgraduate Mathematics Notes: Curvature Mathematical Formula Derivation

1. Curvature of the curve

• The curvature of geometry is defined differently for different objects. Let's first look at the simplest plane curve.
• First, divide the curve into infinitesimal segments, and each segment is regarded as a small arc of a circle. This circle is called the "Osculating Circle". Since it only intersects the curve in a very small section, it is also called "Kissing Circle". The radius of this circle is called the "radius of curvature".
• "Curvature" is a vector that points from a reference point on the arc to the center of the approximation circle. The reciprocal of the radius of curvature of an intimate circle is the "curvature" of the arc at that point. So, the closer the curve is to a straight line, the larger the radius of curvature and the smaller the curvature at that point. Line curvature origin is zero.

2. Representation of the curve

Curves on a 2D plane have two parametric forms, as follows:

• Parametric Equation 1
  
• Parametric equation 2
  
  The above two parametric equations can uniquely determine a curve in a two-dimensional plane. Therefore, the formulas for the curvature, the derivative of the curvature, and the derivative of the derivative of the curvature calculated below have two equivalent forms.

3. Curvature calculation formula and derivation

First give the familiar curvature calculation formula:

and:

3.1 Parametric equation 1 Derivation of curvature formula

3.2 Parametric equation 2 Curvature formula derivation

3.3 Summary

The derivation process of the curvature formula obtained from the two parametric equations is similar, and the final formula form is also similar. When representing a curve, the parametric equations used in different situations are different. For simplicity, the two parametric equations can be unified, and when x ( t ) = t , parametric equation 1 becomes parametric equation 2. At this time, x ′ = 1 , x ′ ′ = 0 , and substituting into Equation (1) yields Equation (2). In the following, only the curvature derivative k' and the derivative k'' of the curvature derivative with respect to the parametric equation 1 are obtained.

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