How to draw a circle using a single hidden variable

Ideas & Philosophy · Published 18 October 2026 Swipe more like this →

In short: Standard functions often fail when describing complex shapes. A circle, for instance, cannot be captured by a single equation in the form of y = f(x). To solve this, mathematicians use a third, hidden variable to trace paths through space.

In traditional calculus, we typically work with functions where one input produces one output, such as y = f(x). While this works for many shapes, it struggles with curves that loop back on themselves. A circle centered at the origin is a prime example: you cannot write its entire circumference as a single function of x. Attempting to solve for y results in two separate formulas, each representing only a portion of the circle, which makes working with the complete shape mathematically cumbersome.

Parametric equations provide an elegant workaround by introducing a third variable, known as a parameter, often denoted as 't'. Instead of defining y directly through x, we define both x and y as functions of this parameter. As 't' varies, each value produces a specific point in space. The collection of all these points forms a parametric curve. This method allows us to describe multidimensional outputs using a one-dimensional input, effectively drawing a curve in space.

To visualize this, imagine dropping a ping pong ball into a large tank of moving water. If we track the ball's location at any specific time 't', the resulting trace of all its positions over time creates a parametric curve. While sketching these curves manually can be difficult—often requiring us to pick arbitrary values for 't' and plot the resulting points—the parametric approach allows us to represent complex, overlapping, or looping paths that standard algebraic equations simply cannot capture.

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