Predicting a Ball's Landing from Three Snapshots

A ball-catching robot's camera sees the ball at three moments. A thrown ball follows a parabola y = ax² + bx + c, and each snapshot gives one equation in the unknowns a, b and c. Solving the three equations together predicts where the ball will land.

How it works

Systems of three linear equations

In one sentence

A system of three linear equations in three unknowns has a unique solution when its coefficient matrix is invertible. For a parabola through $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ with distinct $x_k$ , the matrix with rows $(x_k^2, x_k, 1)$ (a Vandermonde matrix) is invertible, so exactly one quadratic fits.

General formula

\begin{cases} {\htmlData{sym=0}{a}} {\htmlData{sym=3}{x_1}}^2 + {\htmlData{sym=1}{b}} {\htmlData{sym=3}{x_1}} + {\htmlData{sym=2}{c}} = {\htmlData{sym=4}{y_1}} \\ {\htmlData{sym=0}{a}} x_2^2 + {\htmlData{sym=1}{b}} x_2 + {\htmlData{sym=2}{c}} = y_2 \\ {\htmlData{sym=0}{a}} x_3^2 + {\htmlData{sym=1}{b}} x_3 + {\htmlData{sym=2}{c}} = y_3 \end{cases}

Each snapshot pins the curve at one more place. A parabola has three adjustable numbers, so three pins fix it completely.

What the symbols mean

$a$
Curvature coefficient
Sets how sharply the path bends (negative for a falling ball).
$b$
Linear coefficient
With a, sets where the peak is.
$c$
Constant
The height where the path crosses x = 0.
$x_1$
First snapshot, horizontal
How far along the ball was in the first frame.
$y_1$
First snapshot, height
How high the ball was in the first frame.

A thrown ball's height is a quadratic in its horizontal position, $y = ax^2 + bx + c$ , with three unknown coefficients.
Each snapshot $(x_k, y_k)$ becomes one linear equation in $a$ , $b$ and $c$ : the unknowns appear only to the first power.
Three snapshots give three equations. Eliminate one unknown to get two equations in two unknowns, then one in one unknown, then substitute back: the same elimination an augmented matrix organizes.
With $a$ , $b$ and $c$ known, the landing point is where $ax^2 + bx + c = 0$ , from the quadratic formula.

Learn the math

Linear Systems with Three VariablesAlgebra · Lesson 45, Systems of Equations Augmented MatricesAlgebra · Lesson 46, Systems of Equations