Simple harmonic motion is defined as the projection of a uniform circular motion on any diameter of circle of reference.

Thus, simple harmonic motion is very closely linked to circular motion.

Hence, we can geometrically represent the simple harmonic motion in the form of a circle. The particle executes simple harmonic motion from P to Q starting from the point X. The projection of the point P on Y axis is M. As the particle at P completes one complete revolution around the circle, its projection M, moves to and from about the mean position O along the diameter YOY¹

Consider a reference particle, moving on a circle of reference a, with uniform angular velocity ω. Let the particle trace an angle θ in a time t. Hence, ω = θ / t and OM = y.

From the figure, in ΔOPM, sin θ = OM/OP = y/a

Hence, y = a sin θ = a sin ωt

This represents the simple harmonic motion.

Here, we have considered the projection along y axis. If the projection is along x axis, then x = a sin θ = a sin ωt

Similarly, if the particle starts at the point Q, making angle QOX = Ф₀, then the equations for simple harmonic motion can be represented as

y = a sin (θ + Ф₀) = a sin (ωt + Ф₀)