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Chapter 2: Problem 1

Express a rotation through angle \(\alpha\) about the origin as a transformation of (i) polar coordinates, (ii) Cartesian coordinates. If \(f(r, \theta)=0\) is the equation for a curve in polar coordinates, what is the equation for the transformed curve?

Short Answer

Expert verified
Rotate polar angle \(\theta\) to \(\theta + \alpha\); use rotation formulas for Cartesian coordinates.

Step by step solution

01

Understanding Rotation in Polar Coordinates

To rotate a point about the origin in polar coordinates by an angle \( \alpha \), we simply add \( \alpha \) to the angular coordinate \( \theta \). Thus, a point \((r, \theta)\) after rotation becomes \((r, \theta + \alpha)\).
02

Expressing Transformation in Polar Coordinates

Given a curve described by the equation \( f(r, \theta) = 0 \), the transformed curve after rotation can be expressed as \( f(r, \theta + \alpha) = 0 \). This is because each point is rotated by adding \( \alpha \) to its angular coordinate.
03

Understanding Rotation in Cartesian Coordinates

To rotate a point \((x, y)\) in Cartesian coordinates by an angle \(\alpha\) about the origin, we use the rotation transformation formulas:\[ x' = x \cos \alpha - y \sin \alpha \]\[ y' = x \sin \alpha + y \cos \alpha \]These expressions give the new coordinates \((x', y')\) after rotation.
04

Expressing Transformation in Cartesian Coordinates

A curve given by \(y = f(x)\) is transformed by substituting \((x', y')\) from Step 3 into the equation of the curve, yielding the equation of the rotated curve in terms of the new coordinates.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a way of representing points on a plane using two values: the radial distance ( ") from a fixed origin, and the angle ( heta") from a fixed direction. This system is especially useful in scenarios where problems involve rotation or involve angular components.
  • Instead of using a pair of horizontal and vertical distances like in Cartesian coordinates, polar coordinates use an angle and a radius.
  • A point's position is described as (\(r, \theta\)).
  • When rotating in polar coordinates, you change the angle (\(\theta\)), leaving the radius (\(r\)) unchanged.
By adding a rotation angle (\(\alpha\)), you effectively "spin" the point around the origin, changing its angular component to (\(\theta + \alpha\)). This makes polar coordinates highly intuitive for rotations.
Cartesian Coordinates
Cartesian coordinates represent a point's position using two values: the (\(x\)) and (\(y\)) coordinates. Unlike polar coordinates, Cartesian coordinates are based on using perpendicular axes.
  • Each point is described as (\(x, y\)).
  • These coordinates are highly versatile and commonly used for plotting data in two-dimensional space.
While rotating in Cartesian coordinates is a little more complex than simply adding an angle, it remains straightforward using rotation formulas. For a (\(\alpha\)) rotation:
  • (\(x' = x \cos \alpha - y \sin \alpha\))
  • (\(y' = x \sin \alpha + y \cos \alpha\))
These formulas help transform each point, providing the new (\(x'\)) and (\(y'\)) position after rotation.
Rotation Transformation
A rotation transformation moves points around a center point, keeping them equidistant from it. The transformation is defined by an angle (\(\alpha\)) and a center, often the origin, in basic rotations.
  • In polar coordinates, rotating means adding to the angle (\(\theta\)).
  • In Cartesian coordinates, it involves recalculating (\(x, y\)) positions using trigonometric relations.
Rotations keep shapes and curves intact but change their orientation. This implies simply altering the angle or applying the trigonometric adjustments to Cartesian coordinates.Transformations like these show the deep connection between rotation and geometry.
Curve Equation
A curve equation in either coordinate system represents a continuous group of points that meet a specific condition. For polar coordinates, this condition often involves constant relations of (\(r\)) and (\(\theta\)), such as (\(f(r, \theta) = 0\)).
  • For transformations, every point on the curve alters according to the type of transformation applied.
  • In polar coordinates, a curve rotation is expressed as (\(f(r, \theta + \alpha) = 0\)).
  • In Cartesian, it's altered using the rotated points (\(x', y'\)).
This helps in visualizing rotations not just as isolated rotations of points, but as whole structures changing orientation while maintaining their inherent geometric properties.
Coordinate Systems
Coordinate systems are fundamental methods for locating points. The two most prominent are Cartesian and polar coordinates.
  • Cartesian systems use perpendicular axes with (\(x\)) and (\(y\)) values.
  • Polar systems rely on radius (\(r\)) and angle (\(\theta\)).
  • Each system has its unique application context.
While Cartesian coordinates are best for straightforward distance measurements and alignments, polar coordinates shine with rotational issues and situations where angles are significant.Changing between systems involves mathematical conversions, serving various geometric applications effectively. Understanding both systems is crucial for handling geometrical transformations smoothly across different scenarios.

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Most popular questions from this chapter

Describe the symmetry groups of (a) a scalene triangle, (b) an isosceles triangle, (c) a parabola, (d) a parallelogram. (c) a rhombus, (f) a rectangle, (g) an ellipse.

If the sides of a polygon inscribed in a circle are all equal, the polygon is regular.

Express (a) a half-turn, (b) a quarter-turn, as transformations of (i) Cartesian co ordinates, (ii) polar coordinates. (Take the origin to be the center of rotation.)

The number \(2^{n}+1\) is composite whenever \(n\) is not a power of 2 , \- P. L Wantzel, Journal de Mathematiques pures et appliqu茅es, 2 (1837), pp. 366-372.

If a polygon inscribed in a circle has an odd number of vertices, and all its angles are equal, the polygon is regular. (Marcel Riesz.)
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