Specialist Maths Calculus Week 2

Bezier Curves These are graphic curves to enable us to draw curves on computers. They were designed by Pierre Bezier in 1962 so he could create computer designs for creating new car designs for Renault.

Constructing Bezier Curves Start the curve at point S and end it at point E. Introduce two control points C 1 and C 2, and a parameter t where 0≤ t ≤ 1. S E C1C1 C2C2

Constructing Bezier Curves Points F, G and H are located on SC 1, C 1 C 2 and C 2 E respectively, such that SF:FC 1 = t:1-t, C 1 G:GC 2 = t:1-t and C 2 H:HE = t:1-t S E C1C1 C2C2 t 1-t t t F G H

Constructing Bezier Curves Points I and J are located on FG, and GH respectively, such that FI:IG = t:1-t, GJ:JH = t:1-t. S E C1C1 C2C2 t 1-t t t I J t t F G H

Constructing Bezier Curves Finally P(x(t),y(t)) are the points that describe the Bezier curve such that IP:PJ = t:1-t S E C1C1 C2C2 t 1-t t t I J t t F G H t P

Finding the coordinates of P

Using Matricis Note: P is at S when t = 0, and at E when t = 1.

Example 4 (Ex 6C2)

Solution 4

Example 5 (Ex 6C2)

Solution 5

Example 6 (Ex 6C2) S(-3,5) E(-2,4)

Solution 6 S(-3,5) E(-2,4)

Example 7 (Ex 6C2) S(-3,5) E(-2,4)

Solution 7 S(-3,5) E(-2,4)

Example 8 (Ex 6C2)

Solution 8

Parametric Equations of Tangents to Curves

Example 9 (Ex 6C3)

Solution 9

Using Parametric Forms Parametric representation enables us to express the coordinates of any point in terms of one variable “t”. Example P(t 2 -2,2t+1). We need however to be able to convert from parametric form back to Cartesian form.

Example 11 (Ex 6E1) Find the Cartesian equations of the curve with parametric equations

Solution 11 Find the Cartesian equations of the curve with parametric equations

Example 12 (Ex 6E1) Find the Cartesian equations of the curve with parametric equations

Solution 12 Find the Cartesian equations of the curve with parametric equations

Tangents and Normals If P has coordinates (x(t),y(t)), then The slope of the tangent at t=a is given by:

Example 13 (Ex 6E2) Find the equation to the tangent and the normal to the curve with paramedic equations when t = -1:

Solution 13

Example 14 (Ex 6E2) Find the equation of the tangent to the curve with parametric equations x = 3t +7 and y = 8 – 2t 2, having slope –4.

Solution 14

This Week Text pages 216 to 220, 223 to 226 Exercise 6C2 Q Exercise 6C3 Q1, 2 Exercise 6E1 Q 1, 2 Exercise 6E2 Q 1-5