Length, Area, and Curvature in Differential Geometry

Metric Concepts of Differential Geometry

Length as a Fundamental Concept

The length of a curve is the first and most fundamental metric concept. For a curve r(t) = (x(t), y(t), z(t)) on [a, b]:

L = ∫_a^b |r'(t)| dt = ∫_a^b √(x'(t)² + y'(t)² + z'(t)²) dt

Invariance: Length does not depend on parameterization. If we substitute t = φ(u) (φ' > 0): L is preserved—that is the geometric meaning of length.

Variational principle: Among all curves in space between two points, the shortest is the straight segment. This is a minimization problem for the functional L[r] = ∫|r'| dt. On a sphere, the shortest is an arc of a great circle. On a surface—the geodesic.

Numerical example: Spiral r(t) = (cos t, sin t, t) over t ∈ [0, 2π]: r' = (−sin t, cos t, 1). |r'| = √(sin²t + cos²t + 1) = √2. Length: L = 2π√2 ≈ 8.886. For comparison: circle r = 1 gives L = 2π ≈ 6.28. The spiral is longer although it projects onto the same circle.

Curvature and Center of Curvature

Curvature κ(s) = |r''(s)| quantitatively describes "how rapidly" the curve deviates from straight-line motion.

Center of curvature: O(s) = r(s) + (1/κ) ν(s). This is the center of the "best" approximating circle (osculating circle of curvature) at a given point.

Radius of curvature: R = 1/κ. Small R means sharp bend. Large R means gentle bend.

Formula of curvature in general coordinates: κ = |r' × r''|/|r'|³ (without natural parameterization)

For a planar curve y = f(x): κ = |f''(x)|/(1 + f'(x)²)^{3/2}

Example: Parabola y = x²: f' = 2x, f'' = 2. κ(x) = 2/(1+4x²)^{3/2}. Maximum curvature at x = 0: κ₀ = 2 (R = 0.5). At the vertex of the parabola—the greatest bend.

Evolute and Involute

Evolute—the locus of centers of curvature of the curve r(s): E(s) = r(s) + (1/κ(s)) ν(s).

Involute—the reverse concept: a curve for which the given one is the evolute. Geometrically: if you unwind a thread wound around the evolute, the end of the thread describes the involute.

Theorem: The evolute is the envelope of the family of normals to the curve.

Application: gear transmissions. The profile of a gear tooth is the involute of a circle. This ensures constant angular velocity transmission when rotating (constant transmission ratio), which is critically important for precise mechanisms. Invented by Euler in 1765; used in all modern gear reducers.

Isoperimetric Problem

The classical problem: among all closed planar curves with fixed length L, find the one that encompasses the maximum area.

Answer: circle. Area S = L²/(4π), maximal for the circle.

Isoperimetric inequality: L² ≥ 4πS. Equality is achieved only for the circle.

Numerical example: Square with side a: L = 4a, S = a². Check: L² = 16a², 4πS = 4πa² ≈ 12.57a². Indeed, 16 > 12.57 ✓. The square is "inefficient" in area coverage.

Fundamental Theorem of Curves

If continuous functions κ(s) > 0 and χ(s) are given, then there exists a unique (up to motion) curve with these natural equations. This means: all geometrically significant properties of the curve are determined by the pair (κ(s), χ(s)).

Bending of curves: By the fundamental theorem of Frenet curves, κ(s) and χ(s) together uniquely (up to motion) determine the curve. Conversely, deformation of the curve preserving arc length may change κ(s) and χ(s): spiral and circle have the same length but different curvature and torsion. Knots in 3D are topological invariants related to integrals of curvature (crossing number, Kauffman invariant).

Real Application: Computer Graphics

Bézier curves (1962, Renault)—curves controlled by control points. Designers of cars used them for smooth curves of the body. Curvature of the Bézier curve is calculated from the formula κ = |r' × r''|/|r'|³ and visualized as a "curvature comb"—a standard tool in CAD/CAM systems.

G1-continuity: tangents coincide at the junction. G2-continuity: curvature also coincides. G2 guarantees a "smooth" appearance—used in aerospace design.

Length, Area, and Curvature in Engineering and Science

Formulas for length and area are not only abstract mathematics but also engineering tools. Calculation of the surface of heat exchangers, area of aircraft wings, length of medical catheters and pipelines requires accurate formulas of integral geometry. Curvature of the cross section of a beam determines its stiffness in bending according to Navier's formula: stress in the beam is proportional to curvature κ = M/(EI), where M is the bending moment, E is Young's modulus of elasticity, I is the moment of inertia of the section. That is why I-beam profiles are optimal: they maximize the moment of inertia at minimal mass. The clothoid, for which curvature is proportional to arc length, is used in the design of roads and railways: a smooth transition from zero curvature to a constant value ensures passenger comfort and driving safety. Modern autonomous driving systems use trajectory planning algorithms with limited curvature. Managing the shape of optical lenses via surface curvature is the basis of production of glasses, telescopes, and microscopes. In biomechanics, curvature of the spine is analyzed as a measure of deviation from normal lordosis and kyphosis, allowing quantitative assessment of the degree of scoliosis from X-ray images.

Assignment: (a) For the ellipse r(t) = (a cos t, b sin t, 0): find κ(t). For which t is curvature maximal/minimal? (b) Prove the isoperimetric inequality for a regular n-gon: L = na, S = (na²/4)ctg(π/n). How does the estimate for the circle arise as n→∞? (c) For the spatial curve r(t) = (t, t², t³): compute κ(0) and χ(0) via the formulas r' × r''/|r'|³ etc.