Problem 47 - Precalculus: Functions and Graphs

Question

Moire pattem A moiré pattern is formed when two geometrically regular patterns are superimposed. Shown in the figure is a pattern obtained from the family of circles \(x^{2}+y^{2}=n^{2}\) and the family of horizontal lines \(y=m\) for integers \(m\) and \(n\) (a) Show that the points of intersection of the circle \(x^{2}+y^{2}=n^{2}\) and the line \(y=n-1\) lie on a parabola. (b) Work part (a) using the line \(y=n-2\)

Step-by-Step Solution

Answer

The intersection points form the parabolas \(x^2=2y+1\) for part (a) and \(x^2=4y+4\) for part (b).

Step 1: Set up Intersection Equation

For part (a), we have the circle equation \(x^2 + y^2 = n^2\). The line given is \(y = n - 1\). Substitute \(y = n - 1\) into the circle's equation, replacing \(y\) with \(n-1\). This yields the equation \(x^2 + (n-1)^2 = n^2\).

Step 2: Simplify the Intersection Equation

In the equation \(x^2 + (n-1)^2 = n^2\), simplify by expanding \((n-1)^2\) to get \(x^2 + n^2 - 2n + 1 = n^2\). The \(n^2\) terms cancel out on both sides, leaving \(x^2 = 2n - 1\).

Step 3: Express x in terms of y

Coming from \(x^2 = 2n - 1\), find the expression for \(x\): \(x = \pm \sqrt{2n-1}\). This equation represents a set of points \((x, y)\), where \(y\) is fixed at \(n-1\) and \(x\) depends on \(n\).

Step 4: Identify the parabola for part (a)

Since \(y=n-1\) is constant and \(x \in \{\sqrt{2n - 1}, -\sqrt{2n - 1}\}\), these points \((x, n-1)\) satisfy \(x^2 = 2n - 1\). This equation can be rearranged as \(x^2 = 2y + 1\) after realizing \(y = n-1\), which is a parabola opening in the x-direction.

Step 5: Repeat steps for line y=n-2

For part (b), repeat the same procedure using the line \(y = n - 2\). Substitute this into the circle's equation: \(x^2 + (n-2)^2 = n^2\). This simplifies to \(x^2 = 4n - 4\).

Step 6: Identify the parabola for part (b)

With the line \(y = n-2\), from \(x^2 = 4n - 4\), find the parabola by rearranging to \(x^2 = 4y + 4 \). Again, this is a parabola opening in the direction of the x-axis.

Key Concepts

Intersection of Curves

When two different geometric shapes, such as lines and circles, are superimposed on a graph, the points where they meet are referred to as the intersection of curves. Studying these intersections helps to understand their relative positions and behaviors.

In the context of the moiré pattern exercise, we explore intersection points where a straight line intersects with different circles. Specifically, the goal is to find how often these intersections result in another familiar shape, such as a parabola.

To achieve this, apply the values of the line equation into the circle's equation. This will allow you to discern points that fit both equations. After substituting and simplifying, what remains is an equation representing the intersections, and often, you'll find that these points can form distinctly recognized curves or paths.

Circle Equations

Circle equations are fundamental in geometry and represent all points equidistant from a common center. The standard form of a circle's equation in a Cartesian plane is \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

In this exercise, a set of circles sharing the equation \(x^2 + y^2 = n^2\), but potentially differing in size depending on the value of \(n\), is used. These circles centered at the origin expand or shrink based on \(n\), including all points that lie at a distance \(n\) from the center.

When another geometric figure, like a line, intersects these circles, it creates specific points that we can study to understand how such intersections behave, especially when altered by changing the parameters like \(n\) or the line's position.

Parabola

One interesting result of examining intersections of lines and circles is the discovery of parabolic curves. A parabola is a U-shaped curve that can be oriented in different directions based on the equation form. The general equation for a parabola that opens along the x-axis is \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus.

In the provided exercise, after solving for the points of intersection, we see that they lie on a parabola. For instance, by substituting different linear equations into the circle, such as \(y = n-1\) or \(y = n-2\), the resulting simplified equations add up to form parabolas centered along the horizontal direction. Such transformations highlight the beauty and interconnectedness of different shapes in a Cartesian plane.

Recognizing these parabolas helps in visualizing how lines and circles can combine to form new curves, showcasing the dynamism in geometry.

Geometric Transformations

Geometric transformations involve changing the position, size, or shape of a figure within a plane. They include rotations, translations, reflections, and dilations.

In the study of moiré patterns and curve intersections, we often manipulate these shapes mathematically to reveal hidden relationships. By altering parameters within equations, such as shifting a line or altering a circle's radius, we cause transformations.

• Translation: Changing the \(y\) value in equations moves the line either up or down on the graph, changing the intersection points with the circle.
• Scaling: Altering the radius, \(n\), of the circle equation enlarges or reduces the circle's size, impacting how and where it intersects with the line.

Understanding and applying these transformations help underline the significant role geometry plays in math, allowing us to translate algebraic information into visual, easily comprehendible forms. Every transformation can produce surprisingly new and intricate patterns authentically representative of both mathematics and art.