Geometry is the branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, and solids in space.

A circle is a two-dimensional shape that is defined as a set of points that are equidistant from a central point. In other words, a circle is a perfectly round shape that has no corners or edges. The distance between any point on the circle and the central point is called the circle's radius, and it is a constant value that remains the same for every point on the circle. This simple definition of a circle is a fundamental concept in geometry and is used in many areas of mathematics and science.

Trig Function Definitions

Function	Definition
sin(α)	opposite / hypotenuse
cos(α)	adjacent / hypotenuse
tan(α)	opposite / adjacent

Solving Right Triangles

Know	Want	Compute
α, adjacent	opposite	= adjacent * tan(α)
ㅤ	hypotenuse	= adjacent / cos(α)
α, opposite	adjacent	= opposite / tan(α)
ㅤ	hypotenuse	= opposite / sin(α)
α, hypotenuse	adjacent	= hypotenuse * cos(α)
ㅤ	opposite	= hypotenuse * sin(α)

Circle

ABCsincos

Distance

(x1-x0)²+(y1-x0)²=r² If the circle is centred at the origin (0, 0), then the equation simplifies to x²+y²=r²


double
distance(Point2d a, Point2d b)
{
	return sqrt((a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y));
}

Angle

To find angle, in radians, between two points:


double
angle(Point2d a, Point2d b)
{
	return atan2(b.y - a.y, b.x - a.x);
}

Position

To find a point on a circle:


Point2d
PtCircle2d(Point2d c, double r, double angle)
{
	return Pt2d(
		c.x + r * cos(angle),
		c.y + r * sin(angle));
}

Parallel

The slope of the first line is m1=(y2−y1)/(x2−x1) and the slope of the second is m2=(y4−y3)/(x4−x3). The lines are parallel if and only if m1=m2.

Let'?'s say we are given the center point of the circle and its radius. We can now create a loop which iterates from Center.x-Radius to Center.x+Radius or maybe even downwards from Center.x+Radius to Center.x-Radius. Now we have one point on the radius which is the center of the circle and one point which we have the X to, which is located on the circumference. We can then calculate the Y position of this point using the distance formula as in:


Radius = Sqrt ((P1.x - P2.x) ^2 + (P1.y - P2.y) ^2)


cos(x) = 1 - (x^2/2!) + (x^4/4!)...  (An even function)
sin(x) = x -(x^3/3!) + (x^5/5!)....  (An odd function)

add both series together but keep all signs positive and you have

e^x = 1 + x+ (x^2/2!) + (x^3/3!).....

So e^ipi + 1 =0

Tau is the Circle Constant.

Degree Minute Position to Decimal Position


d = M.m / 60
Decimal Degrees = Degrees + .d

Example:
To convert 124° 44.740, a DMS coordinate, to DD.

44.740(m.m) / 60 = 0.74566667
124(degrees) + 0.74566667(.d) = 124.0.74566667

And so 124° 44.740 is 124.0.74566667 in Decimal Degrees.