Glossary

11.1 Polar coordinate system and transformation

3 min read•august 6, 2024

offer a different way to describe points in a plane using distance and . This system is especially useful for circular or spiral shapes. Understanding how to switch between polar and Cartesian coordinates is key for solving certain types of problems.

In this part of the chapter, we'll learn how to use polar coordinates and transform between systems. This knowledge sets us up for tackling double integrals in polar form, which can simplify calculations for circular regions.

Polar Coordinate System

Defining Polar Coordinates

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  • Polar coordinates represent points in a two-dimensional plane using a radius rr and an angle θ\theta
  • The radius rr measures the distance from the origin (pole) to the point
  • The angle θ\theta measures the counterclockwise rotation from the positive x-axis to the line segment connecting the origin to the point
  • Polar coordinates are denoted as (r,θ)(r, \theta) where r≥0r \geq 0 and θ\theta can take any real value (radians or degrees)

Relationship to Cartesian Coordinates

  • Polar coordinates can be converted to Cartesian coordinates (x,y)(x, y) using the following equations:
    • x=rcos⁡(θ)x = r \cos(\theta)
    • y=rsin⁡(θ)y = r \sin(\theta)
  • Conversely, Cartesian coordinates can be converted to polar coordinates using:
    • r=x2+y2r = \sqrt{x^2 + y^2}
    • θ=tan⁡−1(yx)\theta = \tan^{-1}(\frac{y}{x}) (principal value) or θ=tan⁡−1(yx)+π\theta = \tan^{-1}(\frac{y}{x}) + \pi (if x<0x < 0)
  • The origin in polar coordinates is represented as (0,θ)(0, \theta) for any value of θ\theta

Coordinate Transformation

Polar to Cartesian Transformation

  • To transform a point from polar coordinates (r,θ)(r, \theta) to Cartesian coordinates (x,y)(x, y), use the equations:
    • x=rcos⁡(θ)x = r \cos(\theta)
    • y=rsin⁡(θ)y = r \sin(\theta)
  • Example: Transform the polar point (3,π4)(3, \frac{\pi}{4}) to Cartesian coordinates
    • x=3cos⁡(π4)≈2.12x = 3 \cos(\frac{\pi}{4}) \approx 2.12
    • y=3sin⁡(π4)≈2.12y = 3 \sin(\frac{\pi}{4}) \approx 2.12
    • The Cartesian coordinates are approximately (2.12,2.12)(2.12, 2.12)

Cartesian to Polar Transformation

  • To transform a point from Cartesian coordinates (x,y)(x, y) to polar coordinates (r,θ)(r, \theta), use the equations:
    • r=x2+y2r = \sqrt{x^2 + y^2}
    • θ=tan⁡−1(yx)\theta = \tan^{-1}(\frac{y}{x}) (principal value) or θ=tan⁡−1(yx)+π\theta = \tan^{-1}(\frac{y}{x}) + \pi (if x<0x < 0)
  • Example: Transform the Cartesian point (−1,1)(-1, 1) to polar coordinates
    • r=(−1)2+12=2r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}
    • θ=tan⁡−1(1−1)+π≈3π4\theta = \tan^{-1}(\frac{1}{-1}) + \pi \approx \frac{3\pi}{4}
    • The polar coordinates are (2,3π4)(\sqrt{2}, \frac{3\pi}{4})

Graphing in Polar Coordinates

Polar Equations

  • A polar equation is an equation in terms of rr and θ\theta that describes a curve in the polar coordinate system
  • The general form of a polar equation is r=f(θ)r = f(\theta), where ff is a function of θ\theta
  • To graph a polar equation, create a table of values for θ\theta (usually in the interval [0,2π][0, 2\pi] or [0,π][0, \pi]) and calculate the corresponding rr values using the equation

Graphing Techniques

  • To graph a polar equation, plot the points (r,θ)(r, \theta) in the polar coordinate system
  • Connect the plotted points with a smooth curve
  • Identify any symmetries in the graph, such as rotational symmetry or symmetry about the polar axis
  • Example: Graph the polar equation r=2cos⁡(3θ)r = 2 \cos(3\theta)
    • Create a table of values for θ\theta in the interval [0,2π][0, 2\pi] and calculate the corresponding rr values
    • Plot the points (r,θ)(r, \theta) in the polar coordinate system
    • Connect the plotted points with a smooth curve
    • The resulting graph is a three-leaved rose curve with rotational symmetry

Special Polar Graphs

  • Some common polar graphs include:
    • Cardioid: r=a(1+cos⁡(θ))r = a(1 + \cos(\theta)) or r=a(1−cos⁡(θ))r = a(1 - \cos(\theta))
    • : r=acos⁡(nθ)r = a \cos(n\theta) or r=asin⁡(nθ)r = a \sin(n\theta), where nn is an integer
    • Limaçon: r=a+bcos⁡(θ)r = a + b \cos(\theta) or r=a+bsin⁡(θ)r = a + b \sin(\theta)
    • Lemniscate: r2=a2cos⁡(2θ)r^2 = a^2 \cos(2\theta)
  • Recognize the equations and characteristics of these special polar graphs to quickly identify them

Key Terms to Review (17)

Angle: An angle is formed by two rays that share a common endpoint, known as the vertex. Angles are measured in degrees or radians, and they play a crucial role in understanding the relationship between different points in space, especially when working with polar coordinates and integration in polar forms. In polar coordinates, angles help define the position of points in relation to the origin and are essential for transforming rectangular coordinates into polar ones.
Angular Displacement: Angular displacement refers to the change in the angle as an object rotates around a specific point or axis. It measures the angle in radians or degrees between the initial and final position of the object, providing insight into rotational motion. This concept is crucial for understanding how objects behave in a polar coordinate system, where angles and distances from a central point are used to describe their positions.
Area in polar coordinates: Area in polar coordinates refers to the method of calculating the area of a region defined in the polar coordinate system, where points are represented by their distance from a central point and an angle. This method allows for more straightforward integration when dealing with shapes that are more naturally described in polar form, such as circles and spirals. The area is often determined using double integrals, where the area element is expressed in terms of the polar variables, leading to simplified calculations.
Cardioids: Cardioids are heart-shaped curves that can be represented as a specific type of polar graph. They are created by tracing the path of a point on the perimeter of a circle that rolls around another circle of equal radius. Cardioids are important in various mathematical contexts, particularly in polar coordinates, as they demonstrate how transformations can create complex shapes from simple circular motions.
Circles: A circle is a shape consisting of all points in a plane that are at a fixed distance, known as the radius, from a central point called the center. In the context of the polar coordinate system, circles can be represented in a unique way, using polar coordinates which relate the distance from the origin and the angle made with the positive x-axis. This allows for different perspectives on circles, especially in terms of transformations and their equations.
Conversion Formulas: Conversion formulas are mathematical equations used to transform coordinates or expressions from one system to another. They play a crucial role in bridging different coordinate systems, allowing for easier computation and understanding of geometric figures in various contexts, such as transitioning between Cartesian and polar coordinates, or between Cartesian and spherical coordinates.
Coordinate transformation: Coordinate transformation refers to the process of converting coordinates from one system to another, allowing for different perspectives on geometric shapes and mathematical problems. This concept is crucial as it facilitates the transition between various coordinate systems, such as Cartesian, polar, cylindrical, and spherical, which helps in simplifying equations and computations in different contexts.
Lemniscates: Lemniscates are special types of curves that resemble the shape of a figure-eight or an infinity symbol. They can be represented in polar coordinates, where their equations typically involve trigonometric functions and produce fascinating symmetric patterns that are visually captivating. The most famous lemniscate is the Lemniscate of Bernoulli, which is defined by a specific polar equation and has significant implications in various branches of mathematics, including calculus and complex analysis.
Length of Polar Curves: The length of polar curves refers to the measurement of the distance along a curve described in polar coordinates. This concept involves calculating the total distance covered by a curve as it moves through different angles, represented by the polar equations, and is typically found using integral calculus. It emphasizes the transition from Cartesian coordinates to polar coordinates, requiring an understanding of how to manipulate these coordinate systems and apply appropriate formulas to derive lengths.
Polar coordinates: Polar coordinates are a two-dimensional coordinate system that uses the distance from a reference point (the origin) and an angle from a reference direction to uniquely determine the position of a point in the plane. This system is particularly useful for problems involving circular or rotational symmetry, allowing for simpler integration and analysis in certain contexts.
Polar graph: A polar graph is a representation of mathematical functions in the polar coordinate system, where each point on the graph is determined by a distance from the origin and an angle from a reference direction. This type of graph provides a different way to visualize relationships between variables compared to traditional Cartesian graphs, making it especially useful for modeling periodic phenomena and shapes such as circles and spirals.
Polar to rectangular transformation: Polar to rectangular transformation is the process of converting coordinates from the polar system, which uses a radius and an angle, to the rectangular (or Cartesian) system, which uses x and y coordinates. This transformation is essential for visualizing and working with polar equations in a familiar rectangular format, allowing for easier manipulation and analysis of functions and shapes.
R = √(x² + y²): The equation r = √(x² + y²) defines the radial coordinate 'r' in a polar coordinate system, where 'r' represents the distance from the origin to a point in a two-dimensional plane. This relationship connects Cartesian coordinates (x, y) to polar coordinates, allowing for seamless transformation between the two systems. Understanding this equation is crucial for interpreting how points are represented in polar coordinates and enables the analysis of geometric properties and calculations in different contexts.
Rectangular to polar transformation: Rectangular to polar transformation is a mathematical process that converts coordinates from the rectangular (Cartesian) system, defined by (x, y), into the polar coordinate system, defined by (r, \theta). This transformation is significant as it allows for easier representation and analysis of circular and rotational patterns, which are often more complex in rectangular coordinates.
Reference Angle: A reference angle is the smallest angle formed by the terminal side of an angle in standard position and the x-axis. It is always a positive acute angle, ranging from 0° to 90°, and is crucial for determining the sine, cosine, and tangent of angles in different quadrants, particularly when working with polar coordinates and transformations.
Rose curves: Rose curves are mathematical graphs that create petal-like shapes in the polar coordinate system. These curves are defined by equations of the form $$r = a imes ext{cos}(k\theta)$$ or $$r = a imes ext{sin}(k\theta)$$, where 'a' determines the length of the petals and 'k' indicates the number of petals. Depending on whether 'k' is even or odd, the number of petals can vary, showcasing the intricate connection between polar equations and visual patterns.
Spirals: Spirals are curves that emanate from a central point, moving farther away as they revolve around it. In the context of the polar coordinate system, spirals can be expressed using polar equations, where the radius varies with the angle, creating unique and fascinating shapes. These spirals can be classified into different types, such as Archimedean and logarithmic spirals, each having distinct properties and applications.
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