Curvature and Normal Vectors of a Curve

Introduction

This section explores the concepts of curvature and normal vectors for smooth curves in the plane and in space. These ideas are fundamental in understanding how curves bend and how to describe their geometric properties using calculus.

Curvature of a Plane Curve

Definition and Interpretation

• Curvature measures the rate at which the unit tangent vector to a curve changes direction with respect to arc length.

• The standard symbol for curvature is the lower-case Greek letter kappa, \( \kappa \).

• Curvature is a function that assigns a scalar value to each point on the curve, indicating how sharply the curve is turning at that point.

• The curve must be at least class \( \mathcal{C}^2 \) (second derivatives exist and are continuous).

• At points where \( \kappa \) is large, the curve turns sharply; where \( \kappa \) is near zero, the curve turns gently.

Definition: If \( \mathbf{T} \) is the unit tangent vector of a smooth curve, the curvature function is defined as:

Computing Curvature with an Arbitrary Parameter

• Direct computation with respect to arc length \( s \) is often difficult.

• For a curve given by \( \mathbf{r}(t) \), the curvature can be computed as:

where \( v = \left| \frac{d\mathbf{r}}{dt} \right| \) is the speed and \( \mathbf{T} = \frac{\mathbf{v}}{v} \) is the unit tangent vector.

Example 1: Curvature of a Line

• For a straight line \( \mathbf{r}(t) = t\mathbf{v} + \mathbf{C} \), the tangent vector is constant, so \( \frac{d\mathbf{T}}{dt} = 0 \).

• Thus, the curvature is zero everywhere on a straight line.

Example 2: Curvature of a Circle

• For a circle of radius \( a \), \( \mathbf{r}(t) = (a \cos t, a \sin t) \).

• The curvature is constant and equals the reciprocal of the radius:

The Principal Normal Vector

Definition and Properties

• The principal (unit) normal vector \( \mathbf{N} \) is a unit vector orthogonal to the tangent vector, pointing toward the center of curvature (the concave side of the curve).

• It is defined where \( \kappa \neq 0 \).

• \( \mathbf{N} \) can be computed as:

• Alternatively, using any parameter \( t \):

Example: Principal Normal for a Circle

• For \( \mathbf{r}(t) = (a \cos t, a \sin t) \), the principal normal vector is \( \mathbf{N} = (-\cos t, -\sin t) \), always pointing toward the center of the circle.

Circle of Curvature (Osculating Circle)

Definition and Properties

• The circle of curvature (or osculating circle) at a point \( P \) on a curve is the circle that:

  • Has the same tangent as the curve at \( P \)

  • Has the same curvature as the curve at \( P \)

  • Lies on the concave side of the curve

• The radius of curvature is \( \rho = \frac{1}{\kappa} \).

• The center of curvature is the center of the osculating circle.

Example: Osculating Circle for \( y = x^2 \) at the Origin

• For \( y = x^2 \) at the origin, the curvature \( \kappa = 2 \), so the radius of curvature is \( \frac{1}{2} \).

• The center of curvature is at \( (0, 1/2) \).

Curvature and Principal Normal for Space Curves

Formulas for Space Curves

• The formulas for curvature and the principal normal vector are the same as for plane curves:

• For example, the curvature of a helix \( \mathbf{r}(t) = (a \cos t, a \sin t, bt) \) is constant and equals \( \frac{a}{a^2 + b^2} \).

Vertex of a Plane Curve

Definition and Examples

• A vertex of a smooth plane curve is a point where the curvature has a local maximum or minimum.

• For conic sections:

  • Parabola: Maximum curvature at the vertex (axis of symmetry meets the curve).

  • Ellipse: Maximum curvature at endpoints of the major axis, minimum at endpoints of the minor axis.

  • Hyperbola: Maximum curvature where the line containing the foci meets the curve.

Summary of Key Objectives

• Define and calculate the curvature function for a smooth curve in the plane or in space.

• Define and calculate the principal normal (unit) vector along a smooth curve in the plane or in space.

• Find the radius and center of the osculating circle (circle of curvature) at a point on a smooth curve in the plane.