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A detailed tutorial on the Spiral of Archimedes. Step by step tutorial including several visual examples of the Spiral of Archimedes for reference.

The Spiral of Archimedes, also known as the Archimedean Spiral and the arithmetic spiral, is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. It is represented by the equation The variable a controls which direction the spiral turns in, and the variable b controls the distance between successive turnings. We would recognize the Spiral of Archimedes as being an ordinary spiral that is often used to express the unexplained in movies and television.

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A detailed tutorial of the definition of a rhodonea curve. Step by step tutorial including several visual examples of rhodonea curves for reference.

Rhodonea curves, also known as rose curves, are one of the most common patterns to find in the graph of polar coordinates. Rhodonea curves have an easy pattern to follow. A rhodonea curve is formed when you have the equation If k is an odd number, then that is the exact number of “petals” the rhodonea curve will have. If k is an even number, then the rhodonea curve will have twice that many “petals”. There are many different forms and varieties of rhodonea curves.

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A detailed tutorial of the definition of a cissoid. Step by step tutorial including a visual example of the definition of a cissoid for reference.

A cissoid is a curve that  is found from two curves and a point. The point is known as “the pole”. Each definition of a cissoid and instructions on how to find the cissoid are a little different, depending on your source. However, the basic definition of a cissoid is that it is an average point that is relative to the pole and found between the two given curves. Keep in mind that the cissoid is a curve, not a point, but the curve is to be drawn from the point that is found.

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A detailed tutorial on the definition of a hypocycloid. Step by step tutorial including a visual example of the definition of a hypocycloid for reference.

A hypocycloid is not really an equation, or a graph, or any true function. A hypocycloid is simply a representation of the edge of a wheel or other circular item rolling on the inside of a circle to form curves. What is more noticeable than the curves it forms is the shape enclosed by the curves, which is almost like a stretched out diamond. This stretched out shape is the real hypocycloid.

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A detailed tutorial on the definition of an epicycloid. Step by step tutorial including a visual example of the definition of an epicycloid for reference.

An epicycloid is not really an equation, or a graph, or any true function. An epicycloid is simply a representation of the edge of a wheel or other circular item rolling along the edge of a circle to form curves. The curve it forms is really several concave down curves side by side, in a circular pattern.

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A detailed tutorial on the definition of a cardioid. Step by step tutorial including a visual example of a cardioid for reference. The regular cardioid shape is when there is no loop within the circle.

A limaçon is a special kind of curve that is formed when a circle rolls around the outside of a circle of equal radius. The cardioid is a unique type of limaçon, where the point generating the curve lies on the rolling circle, resulting in a cusp. The cardioid is named for its resemblance to a heart.

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A detailed tutorial on the definition of a cycloid. Step  by step tutorial including a visual example of the definition of a cycloid for reference.

A cycloid is not really an equation, or a graph, or any true function. A cycloid is simply a representation of the edge of a wheel or other circular item rolling in a straight line to form curves. The curve it forms is really several concave down curves side by side.