Geomancy Fun, Mathematics

Cones and Conic Sections

[latexpage]

Relevant terminology

Generator = a linear line (green line) that intersects a vertical line called the axis at a fixed point (purple dot) called the vertex 

  • Note: the angle between the generator and the axis is called the vertex angle 
  • Rotating the generator 360° around the vertex and tracing out the rotation into a 3D surface gives a conecone generator

Double right circular cone = a cone whose bases (the top and bottom faces of the cone) are perpendicular to the vertical axis and form a perfect circle cone

  •  Directrix = 2D outline of the base
    • The directrix of a right circular cone is always a perfect circle
  • Nappe = lateral surface of a single circular cone
    • For the diagram above,
      • The upper nappe is outlined in pink
      • The lower nappe is outlined in blue

Overview of Conic Sections

Conic Sections 3D
Source Link
Picture
Source Link

conic section is the 2D curve corresponing to a particular 2D “slice” of a 3D cone

  • The angle at which the 2D plane slices the 3D cone determines the type of conic section we get

Parabolas

General equation:

y - k = a(x - h)^2

x - h = a(y - k)^2

Circles

General equation:

(x-h)^2 + (y-k)^2 = R^2

  • The center of the circle is defined as C:(h,k)
  • The radius of the cirlce is defined as R

Ellipses

General equation:

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Hyperbolas

General equation:

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Sources:

 

 

 

 

 

 

 

 

 

 

 

 

 

5 thoughts on “Cones and Conic Sections”

    1. Thanks! 😀 I definitely will try to add some context to my posts. Right now I’m still in blog “throw up” mode, but hopefully things will start to have more structure as I progress making the website. I’ll make sure to check out your site at some point today 😀

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