Polar and 3D (analysis)

ellipsoid

ellipsoid

Elliptic cone

Elliptic paraboloid

Hyperbolic paraboloid (saddle)

Hyperboloid of 1 sheet

Hyperboloid of 2 sheets

Elliptic cylinder

Hyperbolic cylinder

Parabolic cylinder (taco)

Ellipsoid traces

3 ellipses/circles

Elliptic cone traces

2 lines + ellipse

Elliptic paraboloid traces

2 parabolas + ellipse

Hyperbolic paraboloid/saddle traces

2 parabolas opening opposite ways + hyperbola

Hyperboloid of 1 sheet traces

2 hyperbolas + ellipse

Hyperboloid of 2 sheets traces

2 opposite facing parabolas + ellipse

Ellipsoid graph

Elliptic cone graph

Elliptic paraboloid graph

Hyperbolic paraboloid/saddle graph

Hyperboloid of 1 sheet graph

Hyperboloid of 2 sheets graph

Elliptic cylinder graph

Hyperbolic cylinder graph

Parabolic cylinder/taco graph

Limacons equation

r = a + bcos𝜃

r = a - bcos𝜃

Rose curves equation

r = acosN𝜃

r = asinN𝜃

Circles equation

r = acos𝜃

r = asin𝜃

Lemniscates equation

r² = a²sin(2θ)

r² = a²cos(2θ)

Cardioid graph

Inner loop limacon graph

Dimpled limacon graph

Convex limacon graph

r = acos(2θ)

r = acos(3θ)

r = asin(2θ)

r = asin(5θ)

r = acos(θ)

r = asin(θ)

r² = a²sin(2θ)

r² = a²cos(2θ)

r = a

constant r

𝜃 = a

constant 𝜃

r = 𝜃

Cardiod conditions

a = b

Inner loop limacon conditions

b > a

| a/b | < 1

Dimpled limacon conditions

b < a

1 < | a/b | < 2

Convex limacon conditions

2b ≤ a

How to find the first petal location (rose curve)

θ = pi/2n