Bézier curve

A Bézier curve is created as follows:
1) A set of points \(P_0\), ..., \(P_n\) are chosen (in the example \(n=4\)).
2) A set of points \(Q_0\), ..., \(Q_{n-1}\) are defined by \(Q_i=t P_{i+1}+(1-t) P_i\) (shown in green).
3) A set of points \(R_0\), ..., \(R_{n-2}\) are defined by \(R_i=t Q_{i+1}+(1-t) Q_i\) (shown in blue).
\(n\)) After repeating the process \(n\) times, there will be one point. The Bézier curve is the path traced by this point at \(t\) varies between 0 and 1.

What is the Cartesian equation of the curve formed when:
$$P_0=\left(0,1\right)$$ $$P_1=\left(0,0\right)$$ $$P_2=\left(1,0\right)$$

Show answer & extension

If you enjoyed this puzzle, check out Sunday Afternoon Maths XX,
puzzles about graphs, or a random puzzle.


Show me a random puzzle
 Most recent collections 

Advent calendar 2020

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

List of all puzzles


menace ave bases colouring division clocks algebra dodecagons mean games christmas cryptic clues multiples palindromes median crossnumbers money tiling hexagons books lines digits elections floors irreducible numbers squares rectangles integration triangles functions number products planes unit fractions odd numbers balancing scales sums range integers doubling fractions proportion shapes quadratics arrows probability grids differentiation sequences the only crossnumber dice ellipses coordinates surds circles digital clocks star numbers routes regular shapes means graphs averages percentages indices volume prime numbers wordplay folding tube maps spheres area combinatorics calculus geometry addition crossnumber 2d shapes cards angles factors crosswords square numbers chalkdust crossnumber symmetry trigonometry dates cryptic crossnumbers perfect numbers partitions logic probabilty numbers cube numbers gerrymandering parabolas time sum to infinity factorials 3d shapes people maths quadrilaterals sport remainders triangle numbers polygons coins multiplication square roots chocolate speed dominos advent chess complex numbers perimeter pascal's triangle shape rugby taxicab geometry


Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2021