Gerono Lemniscate

Lately I’ve been doing some game development for fun and was exploring ways to make sprites move in a figure-eight pattern. An online search led me to the Lemniscate of Gerono¹, which is a figure-eight curve named after French mathematician Camille-Christophe Gerono². The term lemniscate describes a curve that produces a continuous figure-eight³. What interested me about the Lemniscate of Gerono was the perceived simplicity; being defined parametrically as⁴:

x = a * np.sin(t)
y = a * np.cos(t) * np.sin(t)

Using python this can be plotted using the following code:

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0,2 * np.pi, 1024)

a = 1
x = a * np.sin(t)
y = a * np.cos(t) * np.sin(t)

plt.plot(x,y)
plt.show()

The result is a figure-eight curve in the range (+- a, +- a/2)⁴.

Lemniscate of Gerono

Given parametric equations only contain multiply, sin and cos I thought it would be a fun exercise to implement the curve in Fixed-Point using look up tables⁵. Implementing the lemniscate in fixed point would allow me to draw the curve on a simple display using a microcontroller.

Fixed point implementation and considerations

I have written before about fixed point but in short it is the technique of representing floating point numbers using integers. The article explains how to generate a fixed point lookup table for sin and cos that uses a uint8_t angle (in the range 0 -> 2PI) as an index. I recommend a look at ARM’s fixed point manual⁶ which has some example code for implementing basic mathematical operations such as add and multiply etc. Like with the microboids⁷ project the display is in the range [-1, 1) using Q15 representation. A given (x, y) point in Q15 can then be converted to an integer pixel co-ordinate for writing to an LCD display.

Assuming I followed the aforementioned article to generate the sin and cos lookup tables, the next requirement is a Q15-Arithmetic Multiply function which can be implemented as follows⁶:

#define MAX_BITS (32)
#define MAX_Q (MAX_BITS >> 1U)

int16_t QMath_Multiply(int16_t a, int16_t b, uint16_t q)
{
    ASSERT(q < MAX_Q);

    int16_t result = (int16_t)(((int32_t)a * (int32_t)b) >> q);
    return result;
}

The lemniscate of Gerono can then be implemented as thus:

static uint8_t t;

int16_t a = 0x7FFF /* ~ = 0.99997 in float */

/* x = a * sin(t) */
int16_t x = QMath_Multiply(a, qsin[t], 15);

/* y = a * sin(t) * cos(t) */
int16_t y = QMath_Multiply(a, qsin[t], 15);
y = QMath_Multiply(y, qcos[t], 15);

t++;

After converting the Q15 (x, y) point to a pixel co-ordinate the end result is a lemniscate curve:

Lemniscate of Gerono in Q15 fixed point

1. Wikipedia - Lemniscate of Gerono link ↩

2. Wikipedia -Camille-Christophe Gerono link ↩

3. Wiktionary - Lemniscate link ↩

4. Wolfram Mathworld - Eight Curve link ↩ ↩²

5. Wikipedia - Lookup table link ↩

6. Fixed Point Arithmetic on the ARM ↩ ↩²

7. microboids link ↩