Natural Equation
A natural equation is an equation which specifies a curve independent of any choice of coordinates or parameterization.The study of natural equations began with the following problem: given two functions of one parameter, find theSpace Curve for which the functions are the Curvature and Torsion.
Euler 
gave an integral solution for plane curves (which always have Torsion 
). Call the Angle between the Tangent line to the curve and thex-Axis 
the Tangential Angle, then
 | (1) |
is the Curvature. Then the equations |  |  | (2) |
 | |  | (3) |
where
is the Torsion, are solved by the curve with parametric equations | |  | (4) |
 | |  | (5) |
The equations
and 
are called the natural (or Intrinsic)equations of the space curve. An equation expressing a plane curve in terms of 
and Radius of Curvature 
(or ) is called a Cesàro Equation, and an equation expressing a plane curvein terms of and is called a Whewell Equation.Among the special planar cases which can be solved in terms of elementary functions are the Circle, LogarithmicSpiral, Circle Involute, and Epicycloid. Enneper showed that each of these is the projection of aHelix on a Conic surface of revolution along the axis of symmetry. The above casescorrespond to the Cylinder, Cone, Paraboloid, and Sphere.
See also Cesàro Equation, Intrinsic Equation, Whewell Equation
References
Cesàro, E. Lezioni di Geometria Intrinseca. Napoli, Italy, 1896. Euler, L. Comment. Acad. Petropolit. 8, 66-85, 1736. Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 111-112, 1993. Melzak, Z. A. Companion to Concrete Mathematics, Vol. 2. New York: Wiley, 1976. Struik, D. J. Lectures on Classical Differential Geometry. New York: Dover, pp. 26-28, 1988.
- Tait Cycle
- Tait Flyping Conjecture
- Tait's Hamiltonian Graph Conjecture
- Tait's Knot Conjectures
- Takagi Fractal Curve
- Take-Away Game
- Takeuchi Function
- TAK Function
- Talbot's Curve
- Talisman Hexagon
- Talisman Square
- Tame Algebra
- Tame Knot
- Tangency Theorem
- Tangent
- Tangent Bifurcation
- Tangent Bundle
- Tangent Developable
- Tangent Hyperbolas Method
- Tangential Angle
- Tangential Triangle
- Tangential Triangle Circumcenter
- Tangent Indicatrix
- Tangent Line
- Tangent Map