Sketch Family of Curves & Calculate Envelope

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In summary, the conversation discusses how to sketch a family of curves and calculate the envelope, if it exists. The solution involves using the determinant of partial derivatives and the enveloping condition, but there may be errors in the calculations. Suggestions for tidying up the equation include using trigonometric identities and graphing the curves.
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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
 
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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!
 

What is the concept behind the Sketch Family of Curves & Calculate Envelope?

The concept behind the Sketch Family of Curves & Calculate Envelope is to visualize and analyze a group of related curves by drawing them on a single graph. The envelope curve is the boundary that encloses all of the curves in the family.

How do you identify the envelope curve?

To identify the envelope curve, you need to first graph all of the curves in the family. Then, find the point where the curves are closest together or overlap. This point will be on the envelope curve. Continue this process for multiple points along the curves to accurately plot the envelope curve.

What is the significance of the envelope curve in a family of curves?

The envelope curve helps to understand the behavior of a family of curves by showing the maximum or minimum values of the curves. It also helps to identify any patterns or relationships between the curves in the family.

Can the envelope curve be calculated mathematically?

Yes, the envelope curve can be calculated mathematically by finding the derivative of each curve in the family and setting them equal to each other. The resulting equation will be the equation of the envelope curve.

How is the concept of the Sketch Family of Curves & Calculate Envelope used in real-world applications?

The concept of the Sketch Family of Curves & Calculate Envelope is used in various fields such as engineering, physics, economics, and finance. It helps in analyzing and predicting the behavior of related variables and optimizing processes.

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