nameVoid said:x=rT-dsinT
y=r-dcosT
trochoid for d<r
integrating this on 0,2pi = 4x[0,pi/2] I am getting 2r^2pi-8rd+d^2pi text is just showing 2r^2pi+d^2pi
A trochoid curve is a cyclical curve that is formed by a point on a circle rolling along a straight line. It is similar to a cycloid, but the center of the circle is not fixed.
The trochoid curve can be calculated using the following parametric equations:
x = (r - d)cos(t) + dcot(t)
y = (r - d)sin(t) - d
where r is the radius of the circle, d is the distance between the center of the circle and the starting point, and t is the angle of rotation.
If d is less than r, the resulting trochoid curve will have a loop. This is because the center of the circle will pass through the starting point, causing the point on the circle to retrace its path.
Yes, the trochoid curve can be calculated with any values of r and d. However, if d is greater than or equal to r, the resulting curve will not have a loop and will just be a cycloid.
The trochoid curve has many practical applications, such as in the design of gears and pulleys, the motion of a bicycle wheel, and the shape of water droplets falling from a faucet. It can also be used in the development of mathematical models for cyclical phenomena, such as the tides or planetary orbits.