The conversation between human and AI assistant.
Is there a geometric definition, simple set of parametric equations, or a function in polar coordinates for the infinity symbol/sideways figure 8?
That's really interesting. Could you explain the derivation of this equation a little more?
Sure. The infinity symbol consists of two loops, one in the upper half-plane and one in the lower half-plane. These two loops are mirror images of each other across the horizontal axis. The polar equation for the upper loop is r = a sin(θ), where 'a' is the size of the loop and 'θ' ranges from 0 to π/2. Similarly, the polar equation for the lower loop is r = -a sin(θ), where 'θ' ranges from π/2 to π. Combining these two equations and simplifying, we get r = a sin(2θ) or r = a cos(2θ).