In our first 3D printing lesson, we extruded graphs of regular functions like $y = f(x)$. But regular functions have a major limitation: they can only produce one $y$-value for each $x$-value (the vertical line test). That means you can never use a standard function to create closed loops, cloverleaves, stars, or heart medallions!

To 3D print real-world objects like medallions, keychains, and decorative shapes, we use parametric equations.

Instead of expressing $y$ directly in terms of $x$, both $x$ and $y$ are computed independently from a third variable called a parameter, usually an angle $t$ (or $\text{deg}$): $$x = x(t)$$ $$y = y(t)$$ As the angle $\text{deg}$ sweeps through a full revolution from $0^\circ$ to $360^\circ$, the $(x, y)$ coordinates trace out a complete closed shape. When passed into print3d(x, y, R), our 3D engine turns the path into a seamless, solid, 3D-printable object!

Here is the core code for generating a 4-leaf clover design:

Move the mouse over a dotted box for more information.

• Polar to Cartesian: Any polar shape with radius $r(\theta)$ is easily converted into $x$ and $y$ using $x = r \cdot \cos(\theta)$ and $y = r \cdot \sin(\theta)$. In Lua, dcos() and dsin() take angles directly in degrees ($0^\circ$ to $360^\circ$).

• Controlling Petals & Lobes: The number multiplying deg inside the cosine determines the number of leaves or lobes! Changing 4 * deg to 3 * deg creates a 3-leaf Irish shamrock, and 5 * deg creates a 5-petal star or flower.

• Seamless 3D Printing: Because the curve starts at $0^\circ$ and returns to $360^\circ$, our 3D engine automatically stitches the start and end into a seamless closed ring with no internal dividing walls.

• All coordinates in x and y are interpreted in millimeters on your 3D printer build plate.