Question
a customer went to a boutique and asks the tailor to design a table cloth in the shape of hypocycloid with the equation r(t)= cos^3 i + sin^3 j . she insists to have laces at every edge of the table cloth. calculate the lenght of laces that the tailor have to buy.
Answers
Steve
Assuming you mean that the edge of the cloth is to be trimmed with lace, you want the arc length of the graph.
ds^2 = (3cos^2(t)sin(t))^2 + (3sin^2(t)cos(t))^2 dt
= 9sin(t)cos(t)(cos(t)+sin(t)) dt
integrate that from 0 to pi/2 and multiply that by 4 to get the entire length of lace.
Or, there are various formulas for general arc lengths of hypocycloids, but they will need you to use the form
x = 1/4 (3cos(t)+cos(3t))
y = 1/4 (3sin(t)-sin(3t))
ds^2 = (3cos^2(t)sin(t))^2 + (3sin^2(t)cos(t))^2 dt
= 9sin(t)cos(t)(cos(t)+sin(t)) dt
integrate that from 0 to pi/2 and multiply that by 4 to get the entire length of lace.
Or, there are various formulas for general arc lengths of hypocycloids, but they will need you to use the form
x = 1/4 (3cos(t)+cos(3t))
y = 1/4 (3sin(t)-sin(3t))