This first chapter will cover how to use the unit circle to define the sine, cosine and tangent. This learning will then be used to create spirals and superforms, which will constitute the structure of our nautilus.


Trigonometric ratio

The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine, cosine, and tangent, shortened to \(\sin\), \(\cos\) and \(\tan\). These are defined for acute angle \(\theta\) illustrated beside.

The word soh•cah•toa helps us remember the definitions of sine, cosine, and tangent. Where the terms opposite, adjacent, and hypotenuse refer to the lengths of the sides.

– Soh•cah•toa

  • \(\sin(\theta) = opposite / hypotenuse\)
  • \(\cos(\theta) = adjacent / hypotenuse\)
  • \(\tan(\theta) = opposite / adjacent\)

Unit circle

The unit circle definition allows us to extend the domain of sine and cosine to all real numbers. The process for determining the sine and cosine of any angle \(\theta\) is as follows : Starting from \((0, 1)\), move along the unit circle in the counterclockwise direction until the angle that is formed between your position, the origin, and the positive \(x\)-\(axis\) is equal to \(\theta\).

– Sine and cosine

  • \(\sin(\theta) = yCoordinate\)
  • \(\cos(\theta) = xCoordinate\)

You are probably used to the idea of measuring angles in degrees. Now, we are going to use a mathematically purer way to describe angles called radians. This way, a unit circle revolution is \(2\pi\) radians (being equal to 360°), similarly, a straight angle is \(\pi\) radians and a right angle is \(\pi / 2\) radians.


Polar coordinates

As shown, a radian equals \(180 / \pi \) degrees. Thus, to convert radians to degrees and vice versa, you can apply the following formulas.

– Degrees/radians conversion

  • \(degrees = radians * 180 / \pi\)
  • \(radians = degrees / 180 * \pi\)

Finally, another useful coordinate system known as polar coordinates describes a point in space as an angle of rotation around the origin and a radius from the origin. Its radius is often denoted by \(r\) and its angle by \(\theta\), shortened to \((r, \theta)\).

– References

Next chapter, we will learn to create spirals and supershapes, to form the shell of our procedural nautilus.


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