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Chapter 6: Problem 72

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

Short Answer

Expert verified
Sketching a plane curve from parametric equations involves plotting points on the plane for various 't' values and then joining these points. The orientation of a curve is the direction in which the curve is followed as 't' increases and is often represented by an arrow.

Step by step solution

01

Understanding Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points that make up a geometric object such as a curve or surface, in terms of a variable, called a parameter. For example, for a curve in the plane, we might have \(x = f(t)\) and \(y = g(t)\), where \(f\) and \(g\) are some functions and \(t\) is the parameter.
02

Sketching Plane Curves

To sketch a plane curve from parametric equations, we first consider the 't' values. We find out values of x and y for various values of t and mark these (x, y) points on the plane. Joining these points gives us the desired curve.
03

Orientation of Curve

Orientation of a curve in a plane refers to the direction in which the curve is traversed as the parameter 't' increases. This is usually indicated by an arrow. If the curve is followed in the same direction of increasing 't', the curve is said to be positively oriented or counterclockwise. If it is followed in the opposite direction, the curve is said to be negatively oriented or clockwise.

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