Problem 39 - Algebra and Trigonometry

Question

Geometric lnvariants Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance \(d(P, Q)\) and \(d\left(P^{\prime}, Q^{\prime}\right)\) where \(P^{\prime}\) and \(Q^{\prime}\) are the images of \(P\) and \(Q\) under a rotation of axes.

Step-by-Step Solution

Answer

Yes, the distance between two points is invariant under rotation, as demonstrated by proving \( d(P, Q) = d(P', Q') \).

Step 1: Define the Distance Formula

The distance between two points \( P = (x_1, y_1) \) and \( Q = (x_2, y_2) \) is given by the formula: \[ d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula is derived from the Pythagorean theorem and calculates the Euclidean distance in a two-dimensional space.

Step 2: Understand Rotation Transformation

Under a rotation by an angle \( \theta \), a point \( (x, y) \) is transformed to \( (x', y') \) using the rotation formulas: \[ x' = x \cos \theta - y \sin \theta \] \[ y' = x \sin \theta + y \cos \theta \] These transformations preserve the structure of the plane but change the coordinates of the points according to the rotation angle.

Step 3: Find the Distance After Rotation

Now, define the positions of \( P' \) and \( Q' \) using the rotation transformations: For \( P' = (x_1', y_1') \): \[ x_1' = x_1 \cos \theta - y_1 \sin \theta \] \[ y_1' = x_1 \sin \theta + y_1 \cos \theta \] For \( Q' = (x_2', y_2') \): \[ x_2' = x_2 \cos \theta - y_2 \sin \theta \] \[ y_2' = x_2 \sin \theta + y_2 \cos \theta \] Now, calculate the distance \( d(P', Q') \):\[ d(P', Q') = \sqrt{(x_2' - x_1')^2 + (y_2' - y_1')^2} \]

Step 4: Prove Distance Invariance

Substitute the expressions of \( x_1' \), \( y_1' \), \( x_2' \), and \( y_2' \) into the distance formula for \( d(P', Q') \):\[ d(P', Q') = \sqrt{((x_2 \cos \theta - y_2 \sin \theta) - (x_1 \cos \theta - y_1 \sin \theta))^2 + ((x_2 \sin \theta + y_2 \cos \theta) - (x_1 \sin \theta + y_1 \cos \theta))^2} \] Simplifying, we retrieve:\[ d(P', Q') = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This proves that \( d(P, Q) = d(P', Q') \), indicating that the distance is invariant under rotation.

Key Concepts

Distance Formula

The distance formula is a fantastic tool that allows us to calculate the distance between two points in a two-dimensional plane. Given two points, say \( P = (x_1, y_1) \) and \( Q = (x_2, y_2) \), the distance \( d(P, Q) \) can be computed using the formula:\[ d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This equation is derived from the Pythagorean theorem. Imagine a right triangle where the points are opposite from each other. The line segment connecting these points serves as the hypotenuse. The formula then calculates the hypotenuse by effectively using the lengths of the other two sides of the triangle.

• Simple Steps: Find the difference in the x-values and y-values separately.
• Square both differences: This ensures all values are positive, mimicking distance.
• Sum and take the square root: Finally, this gives you the exact distance.

By using this formula, you can determine the Euclidean distance directly and efficiently.

Rotation Transformation

When studying geometry, especially with transformations, rotation is a key concept. A rotation transformation involves turning a shape around a certain point known as the center of rotation. It requires us to use a specific angle, \( \theta \), to redefine the location of a point in space.For any point \( (x, y) \), the new coordinates after a rotation about the origin are computed as:

• New x-coordinate: \( x' = x \cos \theta - y \sin \theta \)
• New y-coordinate: \( y' = x \sin \theta + y \cos \theta \)

These equations essentially rotate the point while keeping it "the same" in terms of distance from its original position.

• Preservation: Rotations maintain distances between any two points, ensuring the original shape and size are preserved.
• Uses: Computer graphics, engineering, and more, often use rotation transformations to analyze object positions.

In summary, this transformation allows a shape to be spun around a point without altering its size or shape.

Euclidean Distance

Euclidean distance is a term used to describe the "ordinary" straight-line distance between two points in Euclidean space. Image it as the shortest path you could possibly take between two places on a flat map. Mathematically, this distance is what the distance formula calculates, underpinning it with the simple concepts of geometry. Through using differences between x-coordinates and y-coordinates, we can build the understanding that no matter how you rotate or shift the map, the Euclidean distance remains constant.

• Invariance: As proven in the exercise, this distance stays the same even when points are rotated.
• Applications: Used widely in fields like robotics, where movement along the shortest path is crucial, or in logistics, ensuring efficient road maps.

By providing a consistent measure of distance, Euclidean distance serves as a fundamental component in both theoretical and applied areas of geometry.