heart

). We start with an equation of a circle: (x - 1/2)^2 + y^2 = (1/2)^2 ⇔ x^2 - x + y^2 = 0 ⇔ y^2 = x(1 - x). Plot: https://www.wolframalpha.com/input/?i=y%5E2%20%3D%20x%281%20... With the previous equation, we got both negative and positive values of y, hence a full circle. We want to eliminate the negative values of y, thereby resulting in a semicircle lying above the x-axis. We take square root of both sides: y = sqrt(x(1 - x)). Plot: https://www.wolframalpha.com/input/?i=sqrt%28x%281%20-%20x%2... Now, we want a mirror image of this curve on the left side of the y-axis, i.e., for negative values of x, we want the same values of y that we see in the above plot. We do this by substituting x with |x| in the above equation: y = sqrt(|x|(1 - |x|)). Plot: https://www.wolframalpha.com/input/?i=Re%28sqrt%28abs%28x%29... This completes the top portion of the heart with two semicircles. PART 2 - Plotting the Segments of Two Quartic Curves Now we try to complete the bottom portion of the heart. I didn't start with quartic curves. Instead, I began with parabolas and arrived at quartic curves. We begin with a single parabola: y^2 = 1 - x. Plot: https://www.wolframalpha.com/input/?i=y%5E2+%3D+1+-+x We go through the same drill. We first eliminate negative values of y and then mirror the graph on the left side of the y axis: y = sqrt(1 - |x|). Plot: https://www.wolframalpha.com/input/?i=Re%28sqrt%281%20-%20ab... Let us get a mirror of this plot below the x-axis by multiplying the right hand side by -1: y = -sqrt(1 - |x|). Plot: https://www.wolframalpha.com/input/?i=Re%28-sqrt%281%20-%20a... But I wanted to get a little bit of concave curvature at the bottom of the curves to resemble the hearts in playing cards. I could get this by a little trick I learnt somewhere online: We can vary the power of |x| between 0.5 and 1.0 to get different curvatures, i.e., we can vary r in y = -sqrt(1 - |x|^r) where 0.5 <= r <= 1.0 to get different curvatures. I liked the concave curvature with r = 0.5, so I chose y = -sqrt(1 - sqrt(|x|)). This turns the segments of parabolas into segments of quartic curves. Plot: https://www.wolframalpha.com/input/?i=Re%28-sqrt%281%20-%20s... PART 3 - Plotting the Two Curves Together Let us combine the two equations together. The equations for both curves are: y = sqrt(|x|(1 - |x|)), y = -sqrt(1 - sqrt(|x|)). Plot: https://www.wolframalpha.com/input/?i=%7By%20%3D%20Re%28sqrt... PART 4 - Scaling the Second Curve Finally, in the actual plot I printed, laminated, and presented to my girlfriend, I have multiplied the second equation by 1.5 to stretch the bottommost point of the heart from y = -1 to y = -1.5. Since I used equal scaling (aspect ratio = 1) for both axes and printed it without any scaling to get a high quality print, stretching the bottom of the heart was necessary to make the heart look beautiful and wholesome. Plot: https://github.com/susam/heart/blob/master/heart-2014-02-14....