David's Blog

Living a quiet life in Coquitlam, B.C.

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Location: Coquitlam, British Columbia, Canada

Sunday, April 07, 2013

Conical Sections: A Deeper Look Part 1

This post goes into more details about elliptical sections that are created by taking an inclined cutting plane through a right cone. This topic was briefly investigated in a previous post, Blog Post: Ellipse by Conical Section; here I investigate more deeply the details of ellipses by conical sections.

A 3D image of a cone and cutting plane are presented in the post mentioned above. Just below is an image of a side view of the cut going through the cone:


In this image, several variables are measured with respect to the axis of the cone.
H is the overall height of the cone.
R is the radius of the bottom of the cone.
These two characteristics of the cone are used to define another parameter, the height-to-radius ratio: γ = H/B

α is the angle at which the cutting plane goes through the cone, with respect to the horizontal.

hc is the vertical distance from the tip of the cone to the point where the cutting plane intersects the axis of the cone.
rc is the radius of the cone where the cutting plane intersects the axis of the cone (= hc/γ).

ru is the distance from the cone axis to the upper edge of the cone, measured along the inclined plane.
rl is the distance from the cone axis to the lower edge of the cone, measured along the inclined plane.

Looking more closely at one half of the cross-section, a few more properties of the section can be introduced, as illustrated in the following image:


ro is the distance from the cone axis to the center of the ellipse, measured along the elliptical plane.
hqa is the vertical distance from the tip of the cone to the horizontal plane on which lies the center of the ellipse.
rqa is the horizontal distance from the cone axis to the center of the ellipse.

At this point, enough parameters have been introduced. Now it is time to start defining them.


or



or








a, the length of the semi-major axis of the ellipse, can be found as follows:
ru + rl = 2a

Therefore,



b, the length of the semi-minor axis of the ellipse, is



Some generic relationships between the basic parameters of the cone and the cutting plane can also be defined:




Just out of curiousity, I would like to confirm the value of α for which ru is a minimum. Differentiating the expression for ru above with respect to α, and setting the resulting equation equal to 0, we find that ru is a minimum when α = arctan(1/γ) = π/2 - arctan(γ).
In other words, ru is a minimum when the cutting plane goes through the cone at an angle that is perpendicular to the side of the cone.
A similar examination using rl leads to the same requirement.
 
 

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