Sketch Family of Curves & Calculate Envelope

[4quila]

Homework Statement

Sketch the family of curves given and calculate and draw the envelope should it exist.

Homework Equations

X(t,Lambda) = r(cos(t),sin(t))T + (acos(Lambda),bsin(Lambda))T

The Attempt at a Solution

using the determinant of the partial derivatives wrt to t and lambda and the enveloping condition that this must be equal to zero i have

-rbsin(t)cos(lambda) + absin(lambda)cos(t) = 0 -> tan t = a/b tan(lambda)

-> t = tan^-1(a/b tan(lambda))

Subing in that value of t and using cos(tan^-1(x)) = 1/SQRT(1+x^2) and sin((tan^-1(x)) = x/SQRT(1+x^2)

i get

(r/SQRT(1 + (a/b)^2 *tan^2(lambda)) , r*(a/b)*tan(lambda/SQRT(1 + (a/b)^2 *tan^2(lambda))T + (acos(lambda), bsin(lambda))

Just looking at the family it feels each lambda should trace a point on a ellipse which should then have a circle of radius r centred on it so the envelope should then be an ellipse round the outside and inside of these circles for suitable a b r. That doesn't seem immediately apparent from the equation so far. Can it be tidied further? Is it incorrect? Thanks

 

[mighty2000]

for any help.
Thank you for sharing your solution and thought process. I appreciate your use of the determinant and enveloping condition to solve for the envelope. Your solution seems to be on the right track, however, there may be some errors in your calculations. I would suggest double checking your work and equations to make sure they are correct.

In terms of tidying up the equation, I would suggest using trigonometric identities to simplify the expression. For example, you could use the identity tan(t) = sin(t)/cos(t) to rewrite the equation in terms of sine and cosine. You could also try substituting in values for a, b, and r to see if you can identify a pattern or relationship between the parameters and the resulting envelope.

Another suggestion would be to graph the family of curves and the envelope using a graphing calculator or software. This can help you visualize the curves and potentially identify any errors or areas for improvement in your solution.

I hope this helps and good luck with your further exploration of this problem. Keep up the good work as a scientist!