# Interactives

This is a collection of interactive graphics. All of the images have control points and inputs where things can be dragged and interfaced with.

## Unit Circle

This interactive illustrates the connection between the unit circle and the trigonometric functions cosine, sine, and tangent. Click and drag the two control points to change the angle or press the animate button. The symbol θ represents the current angle corresponding to the right triangle with an adjacent side (x) and an opposite side (y).

## Prime Factorization Tree

This interactive generates the prime factorization tree for the given input number. The tree demonstrates how a number can be decomposed into a unique sequence of prime numbers (highlighted in green). This sequence of numbers is called the prime factorization of a number and is related to the fundamental theorem of arithmetic. Each sequence is unique, meaning that no two numbers will have the same prime factorization. The product of the prime factorization of a number will always be equal to the number. Input is limited to the range [2,1000000].

## Exponents, Logarithms, and Trees

This interactive demonstrates the basic properties of exponential growth using the structure of a tree. The base of the exponent corresponds to the number of branches that sprout at each intersection or “branching factor” of the tree. The exponent corresponds to the number of levels in the tree. This is expressed mathematically as some number b (base) raised to the y power (exponent).

## Polar Coordinate System

This interactive demonstrates the Polar Coordinate System. In the polar coordinates the position of a point is defined by its distance from the origin (radius) and its angle relative to the origin. The angle is measured in radians where one full rotation is equal to Tau (τ) radians, where Tau is a geometric constant approximately equal to 6.28.

## Cartesian Coordinate System

This interactive demonstrates the Cartesian Coordinate System. Every point’s position is defined by two components: the x-component which measures the horizontal distance from the origin and the y-component which measures the vertical distance from the origin.

## Cosine Function

This interactive demonstrates the connection between a right triangle of hypotenuse one and the graph of the cosine function. Click and drag either point to change the angle that describes the right triangle, or change the angle in the input box. It is important to note that the adjacent side of the triangle corresponds to the x component of the point. This is because the circle’s center is at the point (0,0), separate from the graph.

## Calculate Tau (6.28...)

This interactive demonstrates how the geometric constant Tau (τ) can be approximated using two polygons. One polygon is inscribed in the circle, the other is cricumscribed around the circle. Then using the perimeter of the two polygons a lower-bound and upper-bound approximation can be calculated. This is because τ is defined geometrically as the length of any circle’s circumference dvided by the length of its radius.

## Sine Function

This interactive demonstrates the connection between every right triangle of hypotenuse one and the graph of the sine function. Click and drag either point to change the angle that describes the right triangle, or change the angle in the input box.

## Angle

This interactive illustrates the definition of an angle formed by two rays. The point where the two rays meet is called the vertex of the angle. Angles are measured using two unit of measure in math: radians and degrees.

## Angle Between Two Vectors

This interactive demonstrates the angle formed between two vectors.

## Area Between Two Vectors

This interactive demonstrates how to vectors form the shape of a parallelogram. This area of this shape is of particular interest because of its application to computer graphics, approximating the surface area of a shape, and more. To calculate the area, we can use the property of the cross product operator between to vectors: the magnitude of the cross product of two vectors is equal to area between them.

## Circle Defined By Three Points

This interactive demonstrates how a circle is uniquely defined by three points. The general equation of the circle is given in the form of: The values for A, B, C, and D can be solved using the following determinants. Expanded the values are given by: Then this can be converted to the center point form of the equation of the circle. The center point is given by:

## Cubic Bezier Curve

This interactive demonstrates a cubic Bézier curve described by four points. Each of the four points can be dragged to change the shape of the curve.

## Line Defined By Two Points

This interactive demonstrates how to points uniquely define a line.

## Modular Arithmetic

This interactive demonstrates the properties of modular arithmetic on a number-line. The modulus operator, represented as , is the same operation as dividing x by a and returning the remainder of the division. The first control point controls the input number x and the second control point controls the number a. Click and drag either point to see the properties of modular arithmetic.

## Point Where Two Lines Intersect

This interactive demonstrates the point where two lines intersect. The interactive has four blue control points which define the shape of two lines. Click and drag any of the control points to see the point of intersection change. Calculating the Position of the Point The general solution to finding the position of the point of intersection is to find or identify the equations of the lines. Then we can solve for the point by setting the two lines equal to each other.

This interactive demonstrates a quadratic Bézier curve described by three points. Each of the three points can be dragged to change the shape of the curve.

## Unit Circle Angle

This interactive demonstrates how angles can be described using the unit circle.

## Unit Circle Right Triangle

This interactive demonstrates the unit circle is used to describe every right triangle with a hypotenuse of one.

## Vector

This interactive demonstrates a vector. Every vector has a magnitude and direction.