3.4 Curvature and torsion

Vector functions can describe curves in 3D space. Curvature and torsion are key measures of how these curves behave. Curvature shows how sharply a curve bends, while torsion reveals how it twists out of a plane.

These concepts build on earlier topics in vector calculus. They help us understand the geometry of curves more deeply, connecting ideas like derivatives and cross products to real-world shapes and motions.

Curvature and Radius of Curvature

Measuring Curvature

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• Curvature quantifies how much a curve deviates from a straight line at a given point
• Calculated using the formula $\kappa = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}$, where $\mathbf{r}(t)$ is the vector function representing the curve
• Higher curvature values indicate sharper turns or bends in the curve (hairpin turns on a mountain road)
• Curvature is an intrinsic property of the curve and does not depend on the coordinate system used

Radius of Curvature and Curve Types

• Radius of curvature is the reciprocal of curvature, given by $\rho = \frac{1}{\kappa}$
• Represents the radius of the osculating circle, which is the circle that best approximates the curve at a given point (a small section of a roller coaster track)
• Plane curves are curves that lie entirely in a single plane, such as circles, ellipses, and parabolas
• Space curves are curves that do not lie in a single plane and have non-zero curvature and torsion, such as helices and spirals

Torsion and Helices

Torsion of Space Curves

• Torsion measures how much a space curve deviates from a plane curve
• Calculated using the formula $\tau = \frac{(\mathbf{r}'(t) \times \mathbf{r}''(t)) \cdot \mathbf{r}'''(t)}{|\mathbf{r}'(t) \times \mathbf{r}''(t)|^2}$
• Positive torsion indicates the curve is turning counterclockwise as it moves along the curve, while negative torsion indicates clockwise turning
• Torsion is zero for plane curves, as they do not deviate from a single plane

Helices and Their Properties

• A helix is a special type of space curve with constant curvature and constant torsion
• Helices can be right-handed (turning counterclockwise) or left-handed (turning clockwise), depending on the sign of the torsion (spiral staircase, DNA molecule)
• The pitch of a helix is the distance between two corresponding points on adjacent turns of the helix, measured along the axis of the helix
• Helices have applications in various fields, such as physics (electromagnetic waves), biology (protein structures), and engineering (springs and screws)

Serret-Frenet Formulas and Arc Length Parameterization

Serret-Frenet Frame and Formulas

• The Serret-Frenet frame is a moving coordinate system attached to a point on a space curve, consisting of the tangent, normal, and binormal vectors
• Tangent vector $\mathbf{T}(t)$ points in the direction of the curve's velocity and is given by $\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$
• Normal vector $\mathbf{N}(t)$ points in the direction of the curve's acceleration and is given by $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}$
• Binormal vector $\mathbf{B}(t)$ is perpendicular to both the tangent and normal vectors and is given by $\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$
• The Serret-Frenet formulas describe the relationships between the tangent, normal, and binormal vectors and the curvature and torsion of the curve (roller coaster track design)

Arc Length Parameterization

• Arc length parameterization is a way to parameterize a curve using the distance along the curve from a fixed starting point
• The arc length parameter $s$ is given by $s(t) = \int_{t_0}^t |\mathbf{r}'(u)| du$, where $t_0$ is the starting point and $t$ is the endpoint
• Arc length parameterization ensures that the curve is traversed at a constant speed of 1 unit per unit of the parameter $s$
• Useful for simplifying calculations involving curvature and torsion, as the formulas become $\kappa = |\mathbf{r}''(s)|$ and $\tau = \frac{(\mathbf{r}'(s) \times \mathbf{r}''(s)) \cdot \mathbf{r}'''(s)}{|\mathbf{r}'(s) \times \mathbf{r}''(s)|^2}$