Quadratic Bezier Curve Formula: P0, P1, P2, and t

The quadratic Bezier curve formula is

where $P_0$ is the start point, $P_1$ is the single control point, $P_2$ is the end point, and $t$ moves from $0$ to $1$ along the curve. At $t = 0$ the curve is exactly at $P_0$; at $t = 1$ it is exactly at $P_2$; between those values, the changing weights blend the three points into a smooth arc.

How the Formula Weights the Points

The point $B(t)$ is the point on the curve for one value of $t$. The formula works for 2D points such as $(x, y)$, 3D points such as $(x, y, z)$, or any coordinate type where points can be multiplied by numbers and added together. For a 2D curve, compute the $x$ coordinate with the same weights, then compute the $y$ coordinate with the same weights.

The three weights are:

Those weights always add to $1$. That is why the formula behaves like a smooth weighted average of the control points rather than an arbitrary polynomial jump.

The draggable handle is $P_1$. As it moves, the endpoints stay fixed while the curve bends toward the handle. That visible pull is exactly what the middle term $2(1 - t)tP_1$ contributes: it has no weight at $t = 0$ or $t = 1$, but it becomes strongest near the middle of the curve. For the broader geometry behind these handles, the full interactive Bezier curves tutorial shows the same quadratic equation alongside cubic curves, tangent handles, and De Casteljau construction.

What P0, P1, P2, and t Mean

$P_0$ is the start point. The curve touches this point when $t = 0$. In drawing and animation tools, this is the first anchor of the segment.

$P_2$ is the end point. The curve touches this point when $t = 1$. Together, $P_0$ and $P_2$ define the endpoints of the visible segment.

$P_1$ is the control point. It pulls the curve away from the straight line between $P_0$ and $P_2$, but it usually is not on the curve. Moving $P_1$ farther from the endpoint line creates a stronger bend. Moving it closer to that line makes the curve flatter. The curve leaves $P_0$ in the direction of $P_1$ and arrives at $P_2$ from the direction of $P_1$, so this one handle controls both ends of the quadratic segment.

$t$ is the curve parameter. It is not a distance in pixels and it is not always proportional to arc length. It is a normalized progress value from $0$ to $1$ that tells the formula how much weight to assign to each control point.

Matching a specific distance along the curve needs arc length rather than parameter values. The sibling page on the Bezier curve arc length formula derives the exact quadratic length and shows how to reparameterize the curve by distance.