|
|
| (613 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| <center>[[File:DC_stereo.png]]</center>
| |
|
| |
|
|
| |
|
| |
| Using Mathematica, we can produce a 3D rendering of how the sectors for Level 1 would have to interact with a steady angle theta with respect to the beam line, as angle phi is rotated through 360 degrees.
| |
| <center>[[File:PhiCone.png]]</center>
| |
|
| |
|
| |
|
| |
| Looking just at sector 1, we can see that the intersection of level 1 and the cone of constant angle theta forms a conic section.
| |
| <center>[[File:Projection_side_view.png]]</center>
| |
|
| |
|
| |
| <center>[[File:Projection_Rear_view.png]]</center>
| |
|
| |
|
| |
| Following the rules of conic sections we know that the eccentricity of the conic is given by:
| |
|
| |
| <center><math>e=\frac{\sin \beta}{\sin\alpha}</math></center>
| |
|
| |
|
| |
| Where β is the angle of the plane, and α is the slant of the cone.
| |
|
| |
| If the conic is an circle, e=0
| |
|
| |
| If the conic is an parabola, e=1
| |
|
| |
| If the conic is an ellipse, <math>e=\sqrt{1-\frac{b^2}{a^2}}</math>
| |
|
| |
| <center>[[File:Conic_section.png]]</center>
| |
|
| |
|
| |
| For ellipses centered at (h,k):
| |
|
| |
| <center><math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math></center>
| |
|
| |
| where
| |
|
| |
| <center><math>a = major\ radius\ (= \frac{1}{2}\ length\ major\ axis)</math></center>
| |
|
| |
|
| |
|
| |
| <center><math>b = minor\ radius\ (= \frac{1}{2}\ length\ minor\ axis)</math></center>
| |
|
| |
| <center>[[File:Ellipse.png]]</center>
| |
|
| |
| Where
| |
| <center><math>c^2=a^2-b^2</math></center>
| |
|
| |
|
| |
|
| |
| For a parabola:
| |
|
| |
| <center><math>4px = y^2</math></center>
| |
|
| |
| where
| |
|
| |
| p = distance from vertex to focus (or directrix)
| |