Question

Circle and radius of curvature Choose a point \(P\) on a smooth curve \(C\) in the plane. The circle of curvature (or osculating circle) at the point \(P\) is the circle that (a) is tangent to \(C\) at \(P,\) (b) has the same curvature as \(C\) at \(P,\) and ( \(c\) ) lies on the same side of \(C\) as the principal unit normal \(\mathbf{N}\) (see figure). The radius of curvature is the radius of the circle of curvature. Show that the radius of curvature is \(1 / \kappa,\) where \(\kappa\) is the curvature of \(C\) at \(P.\)

Short answer

Question: Prove that the radius of curvature (R) is equal to the reciprocal of the curvature (kappa) at a point on a curve. Answer: The radius of curvature (R) is equal to the reciprocal of the curvature (kappa) at a point on a curve because the curve and the corresponding osculating circle share the same curvature, kappa, and lie on the same side of the curve as the principal unit normal vector. This relationship can be represented mathematically as \(R = \frac{1}{\kappa}\).

Step-by-step solution

Definitions

\ First, let's recall the definitions of the important terms: - Curvature (\(\kappa\)): The measure of how quickly a curve changes direction at a given point and it is calculated by taking the magnitude of the derivative of the unit tangent vector (\(\mathbf{T}\)) concerning arc length (\(s\)) i.e. \(\kappa = \|\frac{d\mathbf{T}}{ds}\|\). - Principal unit normal (\(\mathbf{N}\)): A unit vector perpendicular to the tangent vector at a point on a curve. - Circle of curvature (Osculating circle): The circle that is tangent to the curve at a given point, having the same curvature as the curve at that point, and lying on the same side of the curve as the principal unit normal vector. - Radius of curvature: The radius of the osculating circle at a given point on the curve.

Tangent to the curve

\ Consider a point \(P\) on curve \(C\) in the plane. We know that osculating circle at point \(P\) is tangent to the curve. The tangent to the curve at point \(P\) is represented by the unit tangent vector \(\mathbf{T}.\)

Curvature and the osculating circle

\ The osculating circle at point \(P\) has the same curvature as the curve \(C\) at \(P.\) In other words, both the curve and osculating circle share the same curvature \(\kappa.\) The radius of curvature of curve \(C\) at point \(P\) is the radius of the osculating circle at point \(P.\) Let's denote this radius by \(R.\)

Relation between curvature and radius of curvature

\ To find the relation between the curvature \(\kappa\) and the radius of curvature \(R,\) we can use the fact that the osculating circle at point \(P\) lies on the same side of the curve as the principal unit normal vector \(\mathbf{N}.\) We know that the curvature \(\kappa\) can be defined as the rate of change of the unit tangent vector \(\mathbf{T}\) with respect to the arc length \(s.\) Since the curve and osculating circle share the same curvature and principal unit normal vector \(\mathbf{N},\) we have: $$ \kappa = \frac{1}{R} \| \frac{d\mathbf{T}}{ds} \| $$

Proving the radius of curvature

\ Now, we want to prove that the radius of curvature \(R\) is equal to \(1/\kappa.\) Using the relation we found in step 4, we can rearrange the equation to solve for \(R.\) $$R = \frac{1}{\kappa} $$ Hence, we have successfully shown that the radius of curvature \(R\) is equal to \(1/\kappa\), where \(\kappa\) is the curvature of curve \(C\) at point \(P\).

Key concepts

Curvature

Curvature is a fundamental concept in understanding how a curve behaves at a specific point. Imagine a curve as a road you're driving on, and curvature tells you how sharply you're turning at any given moment. Mathematically, curvature ( \(\kappa\) ) is the magnitude of the derivative of the unit tangent vector ( \(\mathbf{T}\) ) with respect to arc length ( \(s\) ). The formula is \(\kappa = \left\|\frac{d\mathbf{T}}{ds}\right\|\) . This means curvature measures how quickly the direction of the curve is changing. More curvature indicates a tighter turn. In the context of the radius of curvature, \(\kappa\) is instrumental in determining how "bendy" or flat a curve is at a point. A larger \(\kappa\) suggests a tighter circle, while a smaller \(\kappa\) suggests a more gradual bend.

Osculating Circle

The osculating circle, or circle of curvature, is like a perfect "fitting" of a circle to the curve at a particular point. At any point \(P\) on a smooth curve \(C\) , this circle is tangent to the curve, shares the same curvature at that point, and lies on the same side as the principal unit normal vector. Imagine it as the best possible circular "hug" the curve can receive at point \(P\) .

• It is tangent to the curve, meaning it just touches the curve at that specific point.
• It has the same curvature as the curve at that point, indicating the same sharpness of turn.
• Lies on the side of the curve as indicated by the principal unit normal vector.

Understanding the osculating circle helps in visualizing the geometry and dynamics of curves.

Unit Tangent Vector

The unit tangent vector ( \(\mathbf{T}\) ) is your roadmap to where a curve is heading at any point. It is a vector of length one that points in the direction of the curve's path. To find it, you'd differentiate the position vector of a curve with respect to time or another parameter and normalize it.
Essential properties include:

• Indicates the direction of the curve's immediate path.
• Its magnitude is always one, making it a "unit" vector.
• Helps define curvature, as \(\kappa = \left\|\frac{d\mathbf{T}}{ds}\right\|\).

The unit tangent vector plays a crucial role in understanding and calculating curvature, as it provides the necessary information about the direction in which the curve is heading.

Arc Length

Arc length is the distance along a curve between two points. Imagine walking along a winding path – the arc length tells you exactly how far you've traveled, regardless of how you've twisted or turned along the way. To calculate arc length for a curve given by \(f(x)\), you can use the integral formula: \( \int \| \mathbf{r}'(t) \| \, dt \).

Key points about arc length:

• It provides a non-linear measure of distance, allowing us to follow the curve's true path.
• Used as a parameter in defining the curvature, where \(\kappa = \left\|\frac{d\mathbf{T}}{ds}\right\|\).
• Essential in real-life applications ranging from engineering to graphics design.

Understanding arc length is vital when analyzing curves, as it gives a true distance measure, essential for calculating other properties like curvature.