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darksaurian 08-28-2015 08:00 PM
tilted circle, ellipse
 
Say you're looking straight at a circle around the origin on a cartesian plane. Now you tilt the circle 45 degrees around the x-axis. Pretend objects don't diminish in size with distance, like there is no z-axis. I don't know the mathematical terminology. Now the circle looks the same width as it did before along the x-axis but it's shorter top to bottom along the y-axis. Is the shape the same as an ellipse? I mean there's only one ellipse for every length and width. Would the tilted circle with those same dimensions be the exact same shape?

smallpond 08-28-2015 09:55 PM
Yes. The equation of an ellipse aligned with the X-axis is
Code:
(x/a)^2 + (y/b)^2 == 1
That means you are scaling (tilting) the x-axis by 1/a and the y-axis by 1/b. If you undo the tilting you get:

Code:
x' = x/a
y' = y/b
x'^2 + y'^2 == 1
That is the equation of a unit circle.

darksaurian 08-28-2015 10:15 PM
thx

I can't mark threads in general as solved? I guess that says a lot about general. I'm using this saw software. When I cut 45 degree mitres it went too far. Occurred to me later I'm cutting with an ellipse now, not a circle. So to speak. Same width, half the height. So it reaches further. Maybe the software is stopping the center of the blade at the same place as if the blade were vertical.


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