Question

Write the parametric equation of Be'zier curve with two control points.

Answer 1

A Bézier curve is defined by a set of control points P0through Pn, where n is called its order (n = 1 for linear, 2 for quadratic, etc.). The first and last control points are always the end points of the curve; however, the intermediate control points (if any) generally do not lie on the curve. The sums in the following sections are to be understood as affine combinations, the coefficients sum to 1.n

A quadratic Bézier curve is the path traced by the function B(t), given points P0, P1, and P2,

{\displaystyle \mathbf {B} (t)=(1-t)[(1-t)\mathbf {P} _{0}+t\mathbf {P} _{1}]+t[(1-t)\mathbf {P} _{1}+t\mathbf {P} _{2}]{\mbox{ , }}0\leq t\leq 1},

which can be interpreted as the linear interpolant of corresponding points on the linear Bézier curves from P0 to P1 and from P1 to P2 respectively. Rearranging the preceding equation yields:

{\displaystyle \mathbf {B} (t)=(1-t)^{2}\mathbf {P} _{0}+2(1-t)t\mathbf {P} _{1}+t^{2}\mathbf {P} _{2}{\mbox{ , }}0\leq t\leq 1.}

This can be written in a way that highlights the symmetry with respect to P1:

{\displaystyle \mathbf {B} (t)=\mathbf {P} _{1}+(1-t)^{2}(\mathbf {P} _{0}-\mathbf {P} _{1})+t^{2}(\mathbf {P} _{2}-\mathbf {P} _{1}){\mbox{ , }}0\leq t\leq 1.}

Which immediately gives the derivative of the Bézier curve with respect to t:

{\displaystyle \mathbf {B} '(t)=2(1-t)(\mathbf {P} _{1}-\mathbf {P} _{0})+2t(\mathbf {P} _{2}-\mathbf {P} _{1})\,.}

from which it can be concluded that the tangents to the curve at P0 and P2 intersect at P1. As t increases from 0 to 1, the curve departs from P0 in the direction of P1, then bends to arrive at P2 from the direction of P1.

The second derivative of the Bézier curve with respect to t is

{\displaystyle \mathbf {B} ''(t)=2(\mathbf {P} _{2}-2\mathbf {P} _{1}+\mathbf {P} _{0})\,.}

Answer 2

Answer:

The parametric equation for the Bezier curve with two control points is:

B(t)=(1-t)²P₀+2(1-t)P₁+t²P₂     t∈[0,1]

where P₀ and P₁ are the control points.

Explanation:

Parametric equation: It defines a group of quantities as functions of one or more independent variables called parameters.

Independent variables: Variables whose value do not change with the change in other variables.

Bezier curve: It is a parametric curve used in computer graphics and related fields.

The first control point and the last control point will always be the endpoints of the curve.

Therefore, we get the parametric equation for the Bezier curve with two control points is B(t)=(1-t)²P₀+2(1-t)P₁+t²P₂ for all t∈[0,1].

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