All curves except one were derived from plotting from polar parametric equations. The javascript add-on (JVG) which I often use here sees all angles as radian (fractional parts are dismissed). Basically that explains the discontinuity and shading differences. JVG has also rotated some curves from the orientation that I understand the equations were defining; that is explained in the next sentence. Then too, no specific point is the Origin; my origin was the upper left corner extending to x = 400 and vertically down to y = 400 (and that is +ve 400). I did basically consider the origin as being [200,200] and had to offset as needed.
Variously respondents will name the curves different but this is the frame from which I derived them:
Parabola
y = x2
Folium of Descartes
x = 3at/(1+t3)
y = 3a2/(1+t3)
Archimede's Spiral
x = Rcos(θ)
y = Rsin(θ)
Cardioid
x = R(2cos(θ) - cos(2θ))
y = R(2sin(θ) - sin(2θ))
Circle
x = Rcos(θ)
y = Rsin(θ)
Lemniscate This was deliberately presented as an "8" rather than an expected "∞"
x = Rcos(θ)sin(θ)/(1+sin2(θ))
y = Rcos(θ)/(1+sin2(θ))
Cycloid
x = R - θ(A - sin(θ))
y = R*(1 - cos(θ))
Brachistrochone appears as an invert cycloid as a reflection of the Cycloid.
It's "y" component is merely -ve for the similar point in the cycloid.
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