Metric Properties of Surfaces

• Thierry Ciblac authored this document about metric properties of surfaces for the Ecole Nationale Supérieure d'Architecture Paris-Malaquais.
• The plan includes local metric properties of surfaces and classification of surfaces based on their curvature.

Local Metric Properties of Surfaces

• Local metric properties are assessed at a surface point using parametric approaches.
• These properties are independent of how the surface is parameterized.
• Local Metric Properties Include: Tangent plane and normal, normal curvatures, principal directions of curvature, Gaussian curvature, average curvature

Tangent Plane and Normal

• The tangent plane to the surface at point M contains both tangents to the iso-parametric curves that intersect at that point.
• The perpendicular to the tangent plane at M defines the normal to the surface, oriented along the vector N.
• The point M(u₀, v₀) is located at the intersection of the iso-parametric curves C(u, v) and C(u₀, v₀).
• The variables tu and tv represent tangent vectors to the iso-parametric curves.
• Vector N is normal to the surface at M.
• The tangent plane at M is defined for any sufficiently regular surface; any curve on the surface passing through M has a tangent at M that lies on the tangent plane.
• The normal is perpendicular to the tangent plane at M, oriented along vector N.
• Vector N is arbitrarily oriented but consistently points to the same side of the surface.

Normal Curvatures

• Intersection curves are constructed on the surface using planes containing the normal at M
• Since the curves are planar, associating each with an osculating circle allows defining curvature, called normal curvature.
• Normal curvature Kₙₒᵣₘ is the curvature of the intersection curve formed by a plane passing through N.
• Some authors denote curvature as p instead of k.
• Normal curvature has a sign given relative to the direction of N.
• lKₙₒᵣₘl = 1/Rₙₒᵣₘ
• Osculating circles and centers of curvature are associated with each planar section.

Principal Directions of Curvature

• For sufficiently regular surfaces, the values of Kₙ vary between extreme values kξ and kη corresponding to two perpendicular planes.
• Kₑ and Kₙₑ are the principal curvatures, and the associated tangent directions are the principal directions of curvature; their centers of curvature are Oξ and Oη.

Gaussian Curvature K

• Carl Friedrich Gauss (1777-1854)
• Gaussian curvature, also known as total curvature, at a surface point equals the product of the two principal curvatures: K = kξ ⋅ kη.
• The sign of K = kξ ⋅ kη distinguishes three types of surface points: elliptical, parabolic, and hyperbolic.

Elliptical Point

• A surface point is elliptical if K = kξ ⋅ kη > 0.
• Principal curvature centers lie on the same side of the surface, with kξ and kη having the same sign (positive product).
• The surface does not cross the tangent plane.
• If kξ = kη, the point is umbilic, as seen in all points of a sphere.
• Dupin's indicatrix is obtained by plotting √R in the direction of curvature; for an elliptical point, this is an ellipse.
• The ellipse of Dupin's indicatrix closely resembles the intersection of the surface with a plane parallel to and near the tangent plane, up to a homothety.

Parabolic Point

• A surface point is parabolic if K = kξ ⋅ kη = 0.
• One principal curvature is zero, while the other is not.
• A special case occurs when both principal curvatures are zero, resulting in all normal curvatures being zero, called a flat top.

Hyperbolic Point

• A surface point is hyperbolic if K = kξ ⋅ kη < 0.
• The principal curvature centers are on opposite sides of the surface, with kξ and kη having opposite signs (negative product).
• The surface crosses the tangent plane.
• Asymptotic directions are directions where the normal curvature kn = 0.
• Values of kξ and kη have opposite signs; normal curvatures vary continuously between these values as the plane rotates around the normal, passing through zero.
• Two positions correspond to a normal curvature kn equal to zero.
• Tangent directions to zero-curvature normal curves are called asymptotic directions.
• The surface section by a plane parallel to the tangent plane near M closely resembles a hyperbola with asymptotic directions.
• In this case, Dupin's indicatrix consists of two hyperbolas with asymptotic directions.

Types of Points Summary

• Elliptic Point (A): K = kξ ⋅ kη > 0; Principal curvatures are non-zero; centers of curvature are on the same side of the tangent plane; the surface is locally concave on the curvature center's side and convex on the other.
• Parabolic Point (B): K = kξ ⋅ kη = 0; One principal curvature is zero. The corresponding curvature center is at infinity; the surface is locally concave on the curvature center's side and convex on the other.
• Hyperbolic Point (C): K = kξ ⋅ kη < 0; Principal curvatures are non-zero; curvature centers are on opposite sides of the tangent plane; the surface is locally neither concave nor convex; two asymptotic directions align with zero normal curvature directions.

Average Curvature

• Sophie Germain (1776-1831)
• The average curvature C of a surface at a point is the average of the principal curvatures: C = (kξ + kη)/2.
• For an elliptic point, C ≠ 0.
• For a parabolic point, K = kξ ⋅ kη = 0; C ≠ 0 if only one of the principal curvatures is zero.
• For a hyperbolic point, K = kξ ⋅ kη < 0; C can be zero if kξ = −kη.

Classification of Surfaces Based on Curvatures

• Surfaces can have Gaussian curvature points with variable signs (e.g., a torus) or constants.
• Surface Categories Based on Gaussian Curvature Sign: Positive : Synclastic surfaces, double curvature surfaces with the same orientation K > 0, Zero: Simple curvature surfaces, developable surfaces ,Negative: Anticlastic surfaces, double curvature surfaces with opposite orientation K < 0
• Surface Categories Based on Constant Mean Curvature: Minimal Surfaces when C = constant = 0, C = constant ≠ 0 correspond to pneumatic surfaces.

Surfaces with Variable-Sign Gaussian Curvature Points

• These surfaces allow for points that are elliptical, parabolic, and hyperbolic.
• Torus serves as an example.

Gaussian Curvatures of a Torus

• Point A: Elliptical, K = kξ ⋅ kη > 0
• Point B: Parabolic, K = kξ ⋅ kη = 0
• Point C: Hyperbolic, K = kξ ⋅ kη < 0
• One circle limits between the elliptical points and the hyperbolic points.

Synclastic Surfaces: K > 0

• Synclastic surfaces have a positive Gaussian curvature, meaning they have double curvature in the same direction.
• Examples include spheres, ellipsoids, and surfaces of revolution with convex meridians.
• Non-ruled quadrics: ellipsoid of revolution (sphere is a special case), hyperboloid of two sheets, elliptic paraboloid.

Gaussian Curvature of a Sphere

• Constant > 0
• All points on a sphere with radius R are elliptical with equal principal curvatures, making them all umbilics.
• The Gaussian curvature is constant and positive: K = 1/R2.
• The mean curvature is also constant: C = -1/R, where the normal is oriented outward from the sphere.

Gaussian Curvature of an Ellipsoid

• Is positive, K > 0
• An ellipsoid is derived from a sphere through two orthogonal affinities relative to two perpendicular planes.
• The first affinity creates an ellipsoid of revolution, while the second produces the ellipsoid.
• Sections through the three orthogonal symmetry planes are characteristic ellipses.

Developable Surfaces K = 0

• Surfaces with a Gaussian curvature K = 0 have one principal curvature equal to zero
• These surfaces are developable, meaning they can be applied to a plane without tearing or overlapping.
• These surfaces include cones, cylinders, and surfaces generated by tangents to a skew curve.
• These are ruled surfaces with zero Gaussian curvature at every point.
• The tangent plane is constant along a generatrix.

Developable Surfaces: Cones

• Their Gaussian curvature is K = 0 at all points on the surface.
• At each point, the direction of zero principal curvature is the generatrix, whereas the other principal curvature corresponds to the normal curvature in the plane perpendicular to the surface normal.
• The tangent plane remains constant along a generatrix.

Developable Surfaces: Cylinders

• Their Gaussian curvature is K = 0 at all points on the surface.
• At each point, the direction of zero principal curvature is the generatrix, and the other principal curvature corresponds to the normal curvature in the plane perpendicular to the surface normal.
• The tangent plane is constant along a generatrix.

Developable Surfaces: Tangential Surfaces

• Tangential surfaces are ruled surfaces formed by the tangents to any skew curve.
• Tangential surfaces are developable.
• Infinitely close generatrices lie on the same plane because they are tangent to a common curve, so the tangent plane is constant along a generatrix.

Helicoid

• Generated by tangents to a right helix
• The tangents to the helix form a constant angle with the plane perpendicular to its axis and is a surface of equal slopes.

Anticlastic Surfaces K < 0

• Surfaces for which Gaussian curvature is negative are those with double curvature of opposite sign.
• Examples include the hyperbolic paraboloid and surfaces of revolution with a concave meridian.
• Ruled Surfaces include: Hyperboloid of revolution and Paraboloid Hyperbolique.

Minimal Surfaces C = 0

• The surfaces for which the mean curvature C = (kξ + kη)/2 = 0 at all points.
• They have the property of being the surfaces of minimal area supported by a given border and only contain hyperbolic points.
• With C = 0, kξ = −kη, meaning kξ and kη have opposite signs, and K = kξ ⋅ kη < 0.
• Minimal surfaces can be obtained physically using films of soap supported by a given contour.
• Energy minimization maintains the film's equilibrium, equating to minimizing the area of the occupied surface.
• These liquid membranes can only resist traction constraints and undergo uniform stress.

Catenoid of Revolution

• A catenoid of revolution is a minimal surface that can be obtained physically by applying soap to circular contours.

Helicoid Right to Plan Director

• A surface created from a contour with a portion of a right helix and segments can generate this

Surfaces of Constant Mean Slope

• Occur when the slope remains constant relative to a reference plane.
• The tangent plane makes a constant, non-zero angle with the plane, and remains constant along each line of slope.
• The surface is developable by portions.