An Overview of Geometries
[a work in progress, it's completed but has yet to be entirely transcribed into HTML]

Edited by Adrian Dorn

1) Algebraic

py i = S ai jxj

(for j=0 to n)

lxi = S bi jyj

(for j=0 to n)

ai j, bi j : coefficients p, l : proportionality factors

fi (x0,..., xn) = 0

2) Analytic

Conic Sections

x²/a² + y²/b² = 1

eccentricity < 1

x²/a² - y²/b² = 1

eccentricity > 1

y² = 2 · px

eccentricity = 1

*The parabola may be though of as a limiting case between an ellipse (e < 1) and a hyperbola (e > 1). It also may be considered as an elongated ellipse with its center, one focus and vertex all coinciding at infinity. The graph of a quadratic polynomial, y = ax² + bx + c, is also a parabola.*

Ax² + Bxy + Cy² + Dx + Ey + F = 0
(general equation for conic sections)

3) Trigonometry

1 revolution = 4 right angles = 360° = 2Pi radians

COS a = COS b · COS c + SIN b · SIN c · COS a

Area of a Spherical Triangle: A = E · r ²

3a) Analytic Trigonometry

Polar Coordinates: (r, O)
Linear Transformation:
x = r · cos(O), y = r · sin(O)

The polar equation of a circle expresses the fact that
the curve is independent of O and has constant radius.

x = x' + h, y = y' + h

*A combination of dimensions describing a given system forces the centering of the highest degree components in terms of the whole system.*

x = x' · cos(O) - y · sin(O)
y = x' · sin(O) + y' · cos(O)

x'1 = x 1, y' 1 = y 1, z' 1 = kz 1

x'1 = x1, x'2 = x2, x'3 = x3,


Special Curves

  1. Cubical Parabola:
    y = ax3 + bx2 + cx + d
    y = ax2 (x - e)
  2. Semi-cubical Parabola:
    x3 = Ay2
    The curve along which a particle may descend under the action of gravity so as to describe vertical spaces in equal times.
  3. Sinusoidal Spirals:
    rn = an · cos(nO)

  4. A Cycloid is the locus of a point on the circumference of a circle of radius 'r' that rolls without slipping along a straight line: x = r(O - sinO), y = r(1 - cosO). A Cycloid is a kind of Brachistochrone (no, it's not a dinosaur). It is a curve along which a body moves from one point to another under the action of an accelerating force in the least time possible.
  5. A Logarithmic spiral is the curve cutting radius vectors from a fixed point 'x' under a constant angle '*'. The most famous example of which is the "Golden Spiral."
    The form depends on 'c' independent of 'k.' The pole 'x' is an asymptotic point. The pedal of a logarithmic spiral with respect to its pole is a logarithmic spiral, and its evolute is an equal spiral with the same asymptotic point.
  6. A Catenary is the form that a chain assumes when suspended by its ends and acted upon by gravity alone. Euler discovered, in 1744, that a catenary revolved about its asymptote generates a catenoid, the only minimal surface of revolution. The evolute of the tractrix is a catenary. The tractrix revolved about it asymptote generates a pseudosphere, a surface of constant negative curvature.
  7. A Caustic is an envelope of rays emanating from a source.
  8. An Evolute is the envelope of a given curve or, what amounts to the same thing, the locus of the centres of curvature of a given curve. There are an infinite number of such trajectories to a givencurve, and they are said to be parallel curves, i.e., any two cut off equal lengths on common normals.
  9. Isotopic curve: the locus of points of intersection of tangents to a given curve (or a pair of curves) meeting at a constant angle. For a right angle, the isotopic curve is an orthoptic curve. The orthoptic of a parabola is its directrix, of a central conic, its concentric circle; and of a deltoid, a circle.
  10. A Roulette is the envelope of a fixed line situated in the plane of a curve that rolls without slipping on a fixed curve or straight line.
  11. A Tautochrone is a curve down which a particle acted upon by specified forces will descend in the same amount of time from any initial point to the lowest point. When gravity is the force considered, the inverted cycloid with axis vertical has the required property.
  12. A Loxodrome or spherical helix is defined as the curve cutting the meridian of a sphere at a constant angle.
  13. A Helix is the curve cutting the generators of a right circular cylinder under a constant angle. The orthogonal projection of a helix on a plane parallel to the axis of the cylinder is a sine curve.
  14. A skewed or twisted cubic is the locus of the common point of three projective axial pencils of planes.

4) Differential

Equations of a Surface:

  1. xi = xi (U1, U2) for i = 1, 2, 3
    (U1, U2) is a point on the surface.
  2. Vector component form: Ei (U1, U2) for i = 1, 2, 3

X f = S A k (d/d(x k) f

d x i = A i dt, i = 1,...,n

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