# Circle (in the works)

## Description

A circle is a curve such that there is a point C such that the distance from C to any point P on the curve is constant. This point C is called the center of the circle, and the constant distance[C,P] is called the circle's radius.

The polar equation for circle is easily r==C, where C is a constant. The equation for Cartesian coordinate is also easily derived. The distance from two points {x,y} and {c1,c2} is Sqrt[(x-c1)^2 + (y-c2)^2] by the Pythagorean theorem. set this to a constant r, and let x and y be variable, then we have the Cartesian equation Sqrt[(x-c1)^2 + (y-c2)^2] == r

## Formula

Polar formula: r == b, where b is a constant.

Cartesian equation Sqrt[(x-c1)^2 + (y-c2)^2] == b, where b is the constant and {c1,c2} is the center.

Parametric: t*{Cos[t],Sin[t]}, t goes from 0 to 2 Pi.

## Properties

Circles are so fundamental in mathematics that it plays a basic role in almost every branch of geometry or analysis. The following lists some of the geometric properties or notable facts about circles.

### Constant Curvature

By definition, the curvature of a circle is constant everywhere. And, it is simply 1/r, where r is the radius of the circle. So, if the circle is large, its curvature 1/r is small. If the circle is small, 1/r is then large, so the curve is more curved. In extreme cases, if r is 0, we have a circle so small that it is a dot, and its curvature 1/r is infinitly large. Likewise, if r is infinity, the circe is so large that its curvature 1/r is 0, like a line.

### Thales's Theorem

Theorem: Let A and B opposite points on a circle. Let P be a point on a circle. Angle[A,P,B] is a right angle.

Proof: Let O be the circle's center. Let a be the value of angle[O,A,P]. Let b be the value of angle[P,B,O]. Let p be the value of angle[A,P,B]. We want to show that p==Pi/2. Since the angles of a triangle adds to Pi, we have: a + b + p == Pi. Also, note that dist[O,A]==dist[O,P], therefore triangle[A,O,P] is an isosceles triangle, thus angle[A,P,O]==a. Similar argument to the triangle[O,P,B] gets us: angle[O,P,B] == b. Since angle p is the sum of angle APO and OPB, thus we have p = a + b. Combined this equation with a + b + p == Pi, we get p == Pi/2. End of Proof.

#### Converse of Thales's Theorem

Theorem: Let there be a right triangle. Let a circle passes thru the 3 vertexes of the triangle. The longest side of the triangle is a diameter of the circle.

Proof: Let A, B, C be the 3 vertexes of a right triangle, and BC its longest side. Let O be the midpoint of BC. Let D be a line passing O and perpendicular to AB.

It is given that CBA is a right triangle. By construction, CBD is also a right triangle. Because CBA and CBD share the angle at B and are both right triangles, therefore the corresponding side's lengths of the triangles are proportional to a factor. Since O is the midpoint of AC, therefore length BO is half of BC. Further, D is the midpoint of BA. Triangle OBD and ODA both share the right angle at D, and share the side OD, and the distance[B,D]==distance[D,A], therefore the two triangles are congruent, and distance[O,B]==distance[O,A].

Now, using the same argument above, apply the same construction of passing a line thru O and perpendicular to AC, we derive that distance[O,C]==distance[O,A]. Since now distance[O,C]==distance[O,A]==distance[O,B], therefore O is the center of a circle passing ABC.

### Archimedes's Circle

Theorem: Let there be a fixed circle and a fixed point O in the plane. Let P and Q be the intersections of the circle and a line passing O. distance[O,P]*distance[O,Q] is a constant.

Proof: Let the center of the circle denoted by M, and radius be b. Let the distance[O,M] be a.

The triangle[P,M,Q] is a isosceles and O is a point on the line PQ. Let H be the midpoint of PQ. Let h denote distance[M,H] and c denote distance[P,Q]/2. distance[O,P] and distance[O,Q] can be now represented by (OH+c) and (OH-c) if O is outside of PQ, or (OH+c) and (c-OH) if O is inside of PQ. And their product is either (OH^2 - c^2) or (c^2 -OH^2).

Now, OHM is a right triangle with OM := a, so we have OH^2 == a^2-h^2. Similarly, c^2 == b^2 - h^2. Substitute OH^2 and c^2 in the last expression and simplify we have a^2-b^2 if O is outside of PQ, or (b^2-a^2) if O is inside of PQ. Both of which is a constant. If O coincides with P or Q then OP*OQ == 0. End of Proof.

### Circle defined by Three points

Given 3 non-linear points, there is a unique circle passing through them. The center of the circle is the intersection of two bisectors of any two given point. If the 3 points lies on a line, then the “circle” passing them will be a line. This is another point of view that lines can be considered as circles with infinite radius.

Similarly, given a circle and a point not on the plane of the circle, a sphere is defined. Alternatively, a sphere is defined with 4 points in space not all co-planar. The proof can be easily constructed by considering cross-sections as the planar case, and locate the sphere's center by bisection.

### Geometric Inversion and Relation to Lines

A circle is used to define geometric inversion. In this way, a line can be considered as a circle with infinite radius, and reflection over a line is just a special case of geometric inversion.

### Conic Sections

### Pole and Polars

### Ratio of Circumference and Diameter

The ratio of the circumference of a circle and its diameter is a constant, called pi, and has the value approximate to 3.141592. Pi is a irrational number. Pi is one of the most important constant in mathematics. The study of pi spawned vast number of branches of mathematics since the very beginning of math's history.

### Circular Functions

A circle is the essence of the trig function sine. The trig functions: Sin, Cos, Tan, Sec, Cos, Cot are all based on a single function Sin. The essence of the sine function is that we want a smooth function that oscillates, as to form a wave. The natural development of this idea, is to base on a point moving on circle in constant speed, and record its height with respect to time. This is why, the function sine is sometimes called circular function.

### Steiner Circles

See p.184, p.136 of Visual Complex Analysis