Plane and Space Curves Curvature-based Features

Description of Plane Curves Explicit form: Implicit form: Parametric Form:

Arc length is curvilinear distance along curve

Tangent and Normal Vectors. Orientation Unit speed parameterization Local frame

Arc Length Parameterization Unit speed parameterization

Curvature, Curvature Vector The curvature at a point measures the rate of curving (bending) as the point moves along the curve with unit speed When orientation is changed the curvature changes its sign, the curvature vector remains the same Straight line: circle oriented by its inner normal :

Curvature Profile Visualization of curvature vectors helps in curve quality evaluation

Reconstruction Curves from its Curvature

Frenet Formulas

Frenet Formulas

Curvature, Circle of Curvature is tangent to the curve at P has the same curvature as the curve has at P lies toward the concave or inner side of the curve at P Circle of curvature is the best fitted circle at P

Circle of Curvature is Best Fitted A curve A circle Best fitness

Evolute is the locus of centers of curvature Evolute cusps correspond to curvature extrema Evolute has singularities (cusps) Curve normals are tangent to the evolute Proof = Frenet formulas + + Taylor series expansion regular point curvature max > 0 curvature min > 0

Taylor Series Expansion

Families and Envelopes Consider a family of curves.The envelope of the family is a curve which is tangent to every curve from the family It is convenient to define a family of curves in implicit form Where parameterizes the family Then the envelope is given by

Evolute as the Envelope of Normals Evolute is the envelope of normals Optically the evolute of a given curve can be described as the locus of points where the light rays emitted by the the curve in the normal directions are concentrated. At the singularities of the evolute the concentration is even grater.

Evolute = Envelope of Normals. Derivation A curve parameterized by arc length The family of normals (1) (2) For arbitrary point From (1) and (2):

Caustics, coffeecup caustic The envelope of a family of rays is called a caustic. A so-called coffeecup caustic is clearly visible on the inner surface of a cup when sun shines on it. The coffeecup caustic is generated by light rays reflected by the inner cup surface. The coffeecup caustic has a crescent shape

Caustics by Reflection Definition of orthotomic Given a curvilinear mirror and a source of light, the caustic generated by the light rays reflected by the mirror is the evolute of the orthotomic

Optical Properties of Conic Sections

Offset Curves A d-offset curve consists of points d-equidistant from a given curve. Offset curves are important for many engineering applications If d increases, d-offset develops singularities (cusps). Evolute is swept by cusps of the offset curves

Evolute is Locus of Cusps of Offset Curves d-offset is arc length for d-offset is Frenet basis for d-offset Curvature of d-offset becomes infinite when

The evolute of the involute of a given curve is the curve itself If a line rolls without slipping as a tangent along a curve, then the path of a fixed point on the line forms a new curve called involute The evolute of the involute of a given curve is the curve itself

Sea Navigation: How to Determine Longitude Latitude is the angular distance, in degrees, minutes, and seconds of a point north or south of the Equator. Lines of latitude are often referred to as parallels. Longitude is the angular distance, in degrees, minutes, and seconds, of a point east or west of the Prime (Greenwich) Meridian. Lines of longitude are often referred to as meridians.

Sea Navigation: How to Determine Longitude Latitude can be calculated from the position of Polaris Longitude: carry a clock along on board ship set to Greenwich time (for example) at sea, note the time that the sun was at its zenith (i.e. local noon) using this clock compare to the compiled information. Clocks worked by pendulums. For large oscillations (you might expect them at sea) pendulum’s period depends on the swing amplitude.

Involute and Huygens’ Pendulum The involute was introduced by Huygens who used it to improve a pendulum clock mechanism. For a simple pendulum clock, the time of swing of the pendulum is not perfectly constant. It changes slightly if the angle of swing changes. Huygens proposed to change the length of a pendulum in order to make the time of swing of the pendulum the same for all angles. In 1673, Huygens built a clock with a modified pendulum. The top of the pendulum was made of flexible wire which would swing against curved metal cheeks, altering the pendulum's length as it swings

Dual Curves

Medial Axis (Skeleton) The skeleton (medial axis) of a figure is the locus of the centers of circles inside the curve that are bitangent to the boundary of the figure. Skeleton is widely used for shape recognition purposes Skeleton is generated by first self-intersections of offset curves

Skeleton and Evolute Singularities Skeleton endpoints are located at evolute cusps

Animation Zoo

Approximation of the Skeleton by Voronoi Vertices Voronoi diagram of a set of sites is a partition of the plane into regions (Voronoi regions) each of which consists of points closer to one particular site than to any others. Skeleton of a curve can be approximated by the Voronoi diagram of a set of points distributed along the curve

Practical Curvature Estimation Three-Point Curvature Approximations circle approximation angle approximation

Curve Smoothing via Curvature-driven Evolutions Each point of a smooth curve moves in the normal direction with speed equal to the curvature at that point A powerful tool for shape smoothing

Curve evolutions: numerical implementation Explicit Euler method Updating vertex positions

Active Contours in Image Processing A gray-scale image is defined by its intensity function: Curve evolution according to For the limit curve we have along

Active Contours in Image Processing A gray-scale image is defined by its intensity function: Tumor area detection with active contours

Space Curves Frenet frame Frenet formulas

Space Curves: Geometric meaning of Torsion Osculating plane is the best fitted plane Moving along space curve with unit speed Curvature measures the rate of change of the tangent Torsion measures the rate of change of the osculating plane

Local Analysis of Space Curves Taylor series expansion at

Local Analysis of Space Curves Local shape of curve is given by Projection of a space curve onto osculating plane normal plane tangential plane