ANALYTHIC GEOMETRY
OF
THREE DIMENSIONS

Corradi Davide, Codibue Martina5^D liceo scientificoa.s. 2015/2016

Carthesian coordinate system

In a plane the point is originated by two coordinates: x which is associated with the x-axis, an horizontal line with the positive direction to the right, and y, which is associated with the y-axis, i.e. a vertical line with the positive direction upward.

The situation is different when we consider the position of a point in space: in fact we have to consider three planes and, as a consequence, three different axes (x, y, z) and coordinates.

Representation of a point

The distance between two points and the segment

If we consider two different points located in two different positions on the three-dimensional space, the connection between them is defined segment.

The coordinates of the two points are for example:
A(xA,yA,zA) and B(xB,yB,zB)
From these informations we can calculate:
-the lenght of the segment AB:


-its midpoint:

Segment

The plane

Its general equation is:


where a,b and c are coefficients while d is:

Plane

Particular planes

When the equation of the plane is x=0 we obtain the plane created by the y-axis and the z-axis

When the equation is x=k (where k is a number), we obtain a plane which is parallel to the previous one

When the equation of the plane is y=0 we obtain a plane created by the x-axis and the z-axis

When the equation is y=k(where k is a number), we obtain a plane which is parallel to the previous one

When the equation of the plane is z=0 we obtain a plane created by the x-axis and the y-axis

When the equation is z=k(where k is a number), we obtain a plane which is parallel to the previous one

x=0, x=k

y=0, y=k

z=0, z=k

Other informations about planes

The explicit equation of a plane



where:
m= -a/c
n= -b/c
q=

Condition of parallelism between planes:
plane one: ax+by+cz+d=0
plane two: a'x+b'y+c'z+d'=0

Condition of perpendicularity between planes:
plane one: ax+by+cz+d=0
plane two: a'x+b'y+c'z+d'=0

Distance between a point and a plane:

The straight line

A straight line is made of infinite points which derive from the intersection of two different and non parallel planes.
Its general equation can therefore be written as a system of linear equations:

When we have two different points which belong to the straight line we can use another method to find its equation:

Straight line

Definition of surfaces and
the spheric surface

In the three-dimensional space a surface is defined as a set of points called locus whose location satisfies or is determined by one or more particular conditions.
For example these conditions can be an algebric equation.

The spheric surface is defined as a locus whose property is that all its points have the same distance called radius(r) from its center.

Coordinates of the center: C(x0;y0;z0)
Ipothetical point of the sphere: P(x;y;z)


calculating their distance we obtain the equation of the sphere:

Spheric surface

Composition of figures

In this picture we can see a composition of two figures in the three-dimensional space: a spheric surface and a plane.
The peculiarity of this composition is that the plane is tangent to the surface of the sphere, i.e. the two figures shares only one point.

Plane tangent to a spheric surface

Applications of the three-dimensional geometry

When do we use the three-dimensional geometry in real life?
It could be useful for example to represent the ecliptic, i.e. the apparent path of the sun on the celestial sphere.
In this figure the ecliptic is represented during the equinox (vernal or autumnal) by a semicircumference, and here the Sun (point G) goes from East (point C) to West (point B). The segment h1 is the ipothetical ray of the Sun, which arrives to point J, the observer.
This construction can be used to study and calculate the power of a ray of sun which is absorbed by a solar panel located in point J.

Ecliptic