Analytic Foundations of Geometry

2. Parametric Equations

Robert S. Wilson

Definitions

(Vector addition of points) Let   A = (x1, y1)   and let   B = (x2, y2)   be two points. Define

A + B = (x1 + x2, y1 + y2)

(Scalar multiplication of points) Let   A = (x, y)   and let   r   be a real number. Define

rA = (rx, ry)

Let   A   and   B   be two points. The line segment between   A   and   B   is the set of points of the form

C = (1 - t)A + tB

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where

0 < t < 1

A   and   B   are called the endpoints of the line segment between   A   and   B.

The set of points of the form

C = (1 - t)A + tB

where   t > 1   are the points on the other side of   B   from   A,   and the set of such points where   t < 0   are the points on the other side of   A   from   B.

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The set of points of the form

(1 - t)A + tB

where   t > 0   is the set of points on the line determined by   A   and   B   that are on the same side of   A   as   B.

The set of points of the form

(1 - t)A + tB

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where   t < 0   is the set of points on the line determined by   A   and   B   which are on the opposite side of   A  than   B.

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The set of points of the form

(1 - t)A + tB

where   t > 0   is called the ray from   A   through   B.

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Given a line whose equation is

ax + by = c

the set of points whose coordinates satisfy

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ax + by < c

is the set of points one side of the line, and the set of points whose coordinates satisfy

ax + by > c

is the set of points on the other side of the line.

Let   A,   B,   and   C   be three noncolinear points. The angle between   AB   and   AC   is the set of points which are on the same side of   AB   as   C   and the same side of   AC   as   B.   If a point is in the angle between   AB   and   AC,   it is said to be inside of the angle. Points which are not inside of the angle, and not on the rays starting at   A   and going through   B   and   C,   are said to be outside of the angle.

A triangle is a figure determined by three points consisting of the three line segments joining the three points. The three points are called the three vertices, and the line segments between the vertices are called the sides of the triangle.

Let   A,   B,   and   C   be three noncolinear points. The interior of the   triangle ABC   is the set of points which are on the same side of   AB   as   C, the same side of   AC   as   B, and the same side of   BC   as   A. Points in the interior of the triangle are said to be inside of the triangle. Points which are not inside the triangle are said to be outside of the triangle.

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