Curves & Lines
Math   Imaginary math   Trigonometry   Calculus Area & Volume

Line
Slope intercept
y = mx + k
m is slope, Δy/Δx

Point-slope form
y – y₁ = m(x – x₁)

Two points form
y – y₁ = (x – x₁)(y₂ – y₁) / (x₂ – x₁)

Intercept form
(x/a) + (y/b) = 1
a is x intercept
b is y intercept

Standard Form: the standard form of a line is in the form
Ax + By = C where A is a positive integer, and B, and C are integers.

Two lines are perpendicular if the product of their slopes is −1
or one has a slope of 0 (a horizontal line) and the other has an
undefined slope (a vertical line).

distance between two points
d = √(Δx² + Δy² + Δz²)

Distance point to line
line is ax + by + c = 0
point is x₀, y₀
distance is |ax₀ + by₀ + c| / √(a² + b²)

circle
equation of circle with center at a,b and radius r
    (x–a)² + (y–b)² = r²
center at the origin
    x² + y² = r²
Circle A = πr²
length of chord
L = 2√(r²-d²)
    r = radius
    d = perpendicular distance from chord to center
Area of a segment
    A = (r²/2)[(π/180)θ – sinθ]

Plane
General equation of a plane: ax + by + cz + d = 0
if d=0, plane goes thru the origin
if x₀ y₀ z₀ are where plane intersects the 3 axes...
x₀a = y₀b = z₀c = –d
use this to solve for a,b,c,d

sphere
equation of sphere with center at a,b,c and radius r
    (x–a)² + (y–b)² + (z–c)² = r²
center at the origin
    x² + y² + z² = r²

Parabola
Parabola vertical
4p(y–k) = (x–h)²
p is distance vertex to focus, if positive opens up
p is also distance vertex to directorix
vertex at (h, k)

Parabola open sideways
4p(x–h) = (y–k)²
p is distance vertex to focus, if positive opens to the right
p is also distance vertex to directorix
vertex at (h, k)

Ellipse
The area is πab where a and b are one-half of the ellipse's major and minor axes and are the vertices. F₁ and F₂. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis (PF₁ + PF₂ = 2a)

foci are at a distance c from origin along the x axis, where c² = a² – b²

Ellipse whose axis correspond to the x and y axis has the equation
(x/a)² + (y/b)² = 1
a & b are distance from origin to curve along x or y axis.

Ellipse centered at x₁ and y₁, the equation is
((x – x₁)/a)² + ((y – y₁)/b)² = 1
a & b are distance from center to curve along x or y axis.

Hyperbola
The hyperbola consists of the red curves. The asymptotes of the hyperbola are shown as blue dashed lines and intersect at the center of the hyperbola, C. The two focal points are labeled F1 and F2, and the thin black line joining them is the transverse axis. The perpendicular thin black line through the center is the conjugate axis. The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2. The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green). The two vertices are located on the transverse axis at ±a relative to the center. So the parameters are:
a — distance from center C to either vertex
b — length of a segment perpendicular to the transverse axis drawn from each vertex to the asymptotes
c — distance from center C to either Focus point, F1 and F2, and
θ — angle formed by each asymptote with the transverse axis.

The shape of a hyperbola is defined entirely by its eccentricity ε, which is a dimensionless number always greater than one. The distance c from the center to the foci equals aε. The eccentricity can also be defined as the ratio of the distances to either focus and to a corresponding line known as the directrix; hence, the distance from the center to the directrices equals a/ε. In terms of the parameters a, b, c and the angle θ, the eccentricity ε equals
ε = c/a = √(1 + (b²/a²)) = secθ

For example, the eccentricity ε of a rectangular hyperbola
(θ = 45°, a = b) equals √2.

East-West opening hyperbola, ie, the transverse axis of any hyperbola is aligned with the x-axis and is centered on the origin
(x²/a²) – (y²/b²) = 1

North–South opening hyperbola, ie, the transverse axis of any hyperbola is aligned with the y-axis and is centered on the origin
(y²/a²) – (x²/b²) = 1
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