1. Plane and Space Curves Curvature-based Features

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

3. Arc length is curvilinear distance along curve

4. Tangent and Normal Vectors. Orientation
Unit speed parameterization
Local frame

5. Arc Length Parameterization
Unit speed parameterization

6. 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 :

7. Curvature Profile Visualization of curvature vectors
helps in curve quality evaluation

8. Reconstruction Curves from its Curvature

9. Frenet Formulas

10. Frenet Formulas

11. 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

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

13. 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 > curvature min > 0

14. Taylor Series Expansion

15. 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

16. 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.

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

18. 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

19. 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

20. Optical Properties of Conic Sections

21. 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

22. 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

23. 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

24. 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.

25. 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.

26. 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

27. Dual Curves

28. 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

29. Skeleton and Evolute Singularities
Skeleton endpoints are located at evolute cusps

30. Animation Zoo

31. 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

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

33. 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

34. Curve evolutions: numerical implementation
Explicit Euler method
Updating vertex positions

35. 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

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

37. Space Curves
Frenet frame
Frenet formulas

38. 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

39. Local Analysis of Space Curves
Taylor series expansion at

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