Alternative University

Math Preliminaries

Plane Geometry

“The simplest surface is the plane.  The simplest curves are the plane curves, and of these the simplest is the straight line.  The straight line can be defined as the shortest path between two points… The next simplest curve is the circle… We define the circle as the curve whose points are of constant distance from a given point.”
— 
Hilbert & Cohn-Vossen,
Geometry and the Imagination
Figure 1: Circular waves formed by droplet wake.
Points and Lines

In plane geometry, we define a point to be a position on a plane.  For example, if the plane is a map of the world, the city of Vienna would be a point on the plane, the city of Atlanta would be another point on the plane, etc.

A straight line, on a plane, is the set of points connecting the shortest (hence straightest) distance between two points on the plane. Informally, to make a bee line o from one point to another point (the shortest distance for a bee to return to its hive with nectar would be the straightest distance).

Say a push-pin or nail is affixed to each of two points on a plane, then a straight line may be constructed by pulling a string from one nail to the other, like snapping a line o in construction.

Figure 2: Construction worker preparing to snap a chalk line to mark where to cut pavement. [Navy]

In construction, a chalk line is string that is wound up in an inexpensive portable chalk container, which causes colored chalk to attach to the string. The string is then unwound, and affixed and stretched on nails (“staging nails” in Figure 2).

After making the string as tight as possible (which makes it as short as possible, thus straight as possible), it is “snapped” to deposit colored chalk on underlying material in a line showing where to cut the material.

In this case, red chalk will be snapped on asphalt pavement showing where to cut the asphalt. After the string is snapped, it will be removed and the gasoline powered circular saw shown at right will cut along the line of chalk that had been snapped to the pavement.

Circle

A circle is a locus (set of points) that are a constant distance from a given point which is the center of the circle.  That constant distance is referred to as the radius of the circle.  Every point on the circle is that distance (the radial distance) from the center of the circle.

The radius of a circle is like the spokes of a wagon wheel:

Figure 3: Historic wagon wheels.

The length of the perimeter of the circle, referred to as the circumference of the circle, is equal to two times pi times the radius of the circle. Pi is the constant 3.14159…

Denoting the circumference of a circle as 𝐂, the radius of the circle as 𝐫, and denoting pi as 𝛑, the standard equation for circumference of a circle is given:

𝐂  =  𝟐 𝛑 𝐫

This distance (the circumference) may be measured by laying flat the perimeter of a circle, for example laying flat the metal strip from the outer edge of a wagon wheel.

Figure 3 above shows historic wagon wheels with a metal strip attached to the outer edge of each wheel.  You could remove that metal strip (from a wheel) and lay it flat on the ground to measure the circumference of the wheel.

Another way of measuring circumference could be to mark the point of contact on the wheel and road, where the wheel meets the road, then turn the wheel until the mark on the wheel contacts the road again, and measure the distance along the road from the first point of contact to the second point of contact:

Figure 4: Mapping of unit circle circumference to road surface. The distance from one point of contact to the next is 2π r, which is 2π  for r = 1 (a unit circle). [Wikipedia]

For example, you could put a chalk mark on a tire, then measure the distance between successive chalk marks on the road.

Angles

The unit circle is a circle with radius equal to one.  The circumference (2·π·r) of a unit circle is because r = 1.

Figure 5: Unit circle has radius equal to one, and circumference equal to 2π.

In Figure 5 above, the circumference is measured counter-clockwise (CCW ↶), beginning and ending at the East position (3 O’Clock). This is known as a right handed coordinate system.

In this system, half of the circumference (arc length π) ends at the point on the West side of the circle (9 O’Clock), and one-fourth of the circumference (length π ╱ 2) ends at the point on the North side of the circle (12 O’Clock):

Figure 6: Unit circle radians. Center of circle is denoted ‘O’ for Origin.

These perimeter distances (arc lengths along circumference) are referred to as angles, with this unit of measure referred to as radians.

In this coordinate system, measuring of angles starts at the East position. Going all the way around the unit circle (2π radians) returns the position to the East position again (the same position as zero radians). In common language, this is called “going full circle”.

Going one-fourth way around the unit circle (perimeter distance π ⁄ 2) is called a right angle.  Informally, this angle is “just right” for a corner.  In this case, the corner is at the origin.

A line from the origin to the East (zero position), intersecting a line from the origin to the North (π ⁄ 2 position) forms a perfect corner at the origin of the circle. Those two lines are said to be perpendicular or orthogonal. Such an angle is denoted with a square inside the corner:

Figure 7: Unit circle right angle. For a right angle, the distance along the perimeter of a unit circle is π ⁄ 2.

Angles may be measured informally as degrees, where 360 degrees equals 2π.  In that case, a right angle (π ⁄ 2) is 90 degrees. Math formulas, however, always use radians instead of degrees.

Degrees are just a shorthand way of specifying angles, which are converted to radians when any calculations need to be performed, because radians are an actual distance (along the perimeter of a unit circle) that is used in math formulas.

Coordinate Handedness

The coordinate system we have been using is called a right handed coordinate system. In this system, angles are measured counter-clockwise (CCW) from the East direction:

Figure 8: Right handed coordinate system measures angles CCW from East.

This is called a right handed coordinate system, because if you form the hitch hiking gesture with your right hand, and position the hand over Figure 8, with thumb pointing toward you, the rest of the fingers will curl in the direction of angles being measured CCW.

A left handed coordinate system, on the other hand, measures angles in the clockwise (CW) direction, from North:

Figure 9: Left handed coordinate system measures angles CW from North.

Try positioning your left hand, with hitch hiking gesture, and thumb pointing toward you, over that drawing, to see the other fingers of that hand curling in the CW direction.

Left handed coordinate systems are used in fields like cartography. For example, here is the dial of an analog mountaineering compass, showing angles measured CW from North:

Figure 10: Mountaineering compass. Azimuth ring measures angles CW from North. [Arpingstone]

Introductory mathematics in Western countries uses the right handed coordinate system. We will follow that practice in these introductory pages. You will eventually use both coordinate systems. We briefly mentioned left handed coordinate systems now in case you hear of such systems before we cover that later.


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2020–Oct–26  05:12  UTC