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Equation to graph a 180 degree curve comprised of a radius and an ellipse 
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#1
Nov3012, 02:30 PM

#2
Nov3012, 03:46 PM

Mentor
P: 11,602

Some helpful equations: An ellipse aligned with the coordinate axes can be expressed as ##\frac{x^2}{a^2}+\frac{y^2}{b^2}=1## or ##y(x)=\pm b \sqrt{1\frac{x^2}{b^2}}## Curve radius is given by $$R = \frac{(1+y'^2)^{\frac{3}{2}}}{y''}$$ where ' is the derivative with respect to x. The curve radius of a circle is the regular radius of the circle, of course. 45° away from the intersections of the major axes and the ellipse are points with y'=±1. This simplifies curvature to ##R=\frac{\sqrt{8}}{y''}##, but it applies to a special case of your construction only. 


#3
Nov3012, 04:09 PM

P: 46

Thanks, you are right, I should have included that. I will add it, and drawing to my original question/post. The end of the ellipse portion (at the end of the 180 degree curve) would correspond to the middle of the major axis (where the ellipse intersects line B on the drawing, above). Thanks. 


#4
Dec112, 09:28 AM

P: 46

Equation to graph a 180 degree curve comprised of a radius and an ellipse
(below)



#5
Dec112, 12:57 PM

Mentor
P: 11,602

If you know a and b and the 45°part of the ellipse is like in the sketch, with the origin of the coordinate system in its center: Derive y and solve for y'=1 (it has a nice general solution) to find the corresponding xvalue of point P where ellipse and circle meet. Use this xvalue in y(x) to get the corresponding yvalue for P. Use some tool of your choice to draw y(x) between the calculated xvalue and x=a. Derive y' to get y'', insert your calculated xvalue and plug that in the formula for the curve radius to get the curve radius R. The center of your circle is now ##\frac{R}{\sqrt{2}}## to the left and below P. Its center is M(c,d) and the circle equation is ##(xc)^2 + (yd)^2 = R^2## or ##y(x)=d+\sqrt{R^2(xc)^2}##. Draw this for x=cR to point P. 


#6
Dec112, 02:14 PM

P: 46

I'm probably misunderstanding you, but, from P to the center of the circular portion (as opposed to the ellipse portion) should just be R (the radius), not R / sqr 2. Correct? 


#7
Dec212, 07:48 AM

Mentor
P: 11,602

The total distance is R, but it is composed of the distance in x and ydirection, which have the same magnitude.



#8
Dec212, 08:59 AM

P: 46

Can anyone make a drawing or a graph of this, please?
I know this is a math forum, and I do appreciate math, but I am not very good at it. That's why I am here. I don't know how to "derive" things, so I need a more basic explanation. Pretend I'm an average high school freshman and you'd probably be pretty close to my level of understanding. I get that R / sqr 2 = the length of the x and y paths from P to M, now. Thanks 


#9
Dec312, 12:06 PM

P: 46

Anyone?



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