Tuesday, January 12, 2016

Unizor - Geometry3D - Cartesian Coordinates

The purpose of having any coordinates in space is to represent a
position of a point numerically as a sequence of real numbers. In
three-dimensional space this requires a sequence of three real numbers.

Using this representation, we can describe relationship among many
points in space (say, points lying on some plane or on a surface of some
sphere) algebraically.

This approach is fully utilized in a subject of Analytical Geometry that is studied in universities.

Cartesian Coordinates in Space

Cartesian coordinates in space are similar to Cartesian coordinates on a plane and just need an extension to a third dimension.

In any system of coordinates, first of all, we need a point of origin O - some fixed point in space relatively to which we describe a position of any other point.

Next let's define axes (plural of axis) of coordinates.

I think, it's more natural to define first an axis that differentiates
three-dimensional geometry from two-dimensional one on a plane. So,
let's choose a line going through the point of origin O, call it Z-axis and choose a particular direction on this line as positive.

We can conditionally call this Z-axis a vertical axis because it is usually drawn vertically on pictures and illustrations.

Consider now a plane going through the point of origin O perpendicular to Z-axis. This plane will contain two other axes - X-axis and Y-axis - and we will call this plane XY-plane.
We can choose X-axis first within this plane as we wish, then Y-axis
would be a line within this same plane and perpendicular to X-axis.

Both X-axis and Y-axis require a positive direction. Traditionally, we
choose a direction of X-axis as we wish and then choose a direction of
Y-axis in such a way that, looking from the positive ray of Z-axis,
positive ray of X-axis can be rotated counterclockwise to coincide with a
positive direction of Y-axis.

Similar construction could be done if we start from XY-plane, choose
X-axis and Y-axis within it and construct Z-axis as a perpendicular to

Alternatively, we can start with only X-axis, then choose Y-axis
perpendicular to it, thus defining XY-plane, and then construct Z-axis.

Similar to XY-plane, we can define XZ-plane that contains X-axis and Z-axis or YZ-plane that contains Y-axis and Z-axis.

All these methods lead to similar result of having three axes of coordinates, X-axis (denoted as x), Y-axis (denoted as y and Z-axis (denoted as z), mutually perpendicular to each other.

Finally, we have to choose a unit of measurement to be able to convert
geometrical position of any point in space into three coordinates
relative to three axes defined above.

Assume now that we have a point A somewhere in space,
where three axes of coordinates are defined as described above. Our task
is to convert a geometrical position of this point to a sequence of
three real numbers that uniquely represent it. In other words, we will
put into one-to-one correspondence a set of all points in
three-dimensional space to a set of all triplets of real numbers.

The simple way to do it is to draw three plains through this point A, each perpendicular to a corresponding axis (a unique construction):

plain α⊥axis x (let αx=Ax)

plain β⊥axis y (let βy=Ay)

plain γ⊥axis z (let γz=Az)

The above defines three projections of point A onto each axis - Ax, Ay and Az, uniquely defined by a chosen position of point A.

Having a unit of measurement, we can associate the length of segments OAx, OAy and OAz
with real numbers - their length in chosen unit, and assign a sign to
this number - positive if corresponding projection point lies towards
the positive direction of a corresponding axis or negative in an
opposite case.

The above procedure associates a triplet of real numbers with a position of a point in three-dimensional space.

These three numbers have proper names:

signed length of segment OAx is called x-coordinate or abscissa;

signed length of segment OAy is called y-coordinate or ordinate;

signed length of segment OAz is called z-coordinate or applicate.

Let's do an inverse operation to associate a point with a triplet of real numbers.

Obviously, three real numbers can be easily converted into three points
on our three axes by constructing three segments along them - OAx, OAy and OAz. Then we construct three planes through points Ax, Ay and Az correspondingly perpendicular to axes they belong to.

These three planes intersect in one and only one point A, which is the point associated with given triplet of real numbers.

It would be a useful exercise to prove that our relationship between
points in three-dimensional space and triplets of real numbers is indeed
one-to-one correspondence. Rigorous proof of this is very much related
with foundations of geometry and certain axioms that are beyond the
scope of this course. Intuitively, however, it is as obvious as the fact
that for each real number r we can find a point R on a straight line that is on a distance equal to this number (using some unit of measurement) from a certain fixed point O, that is OR=r (for positive r point R is on one side from point O, for negative R - on the other).

As an exercise, here are a few examples of using coordinates to express the geometrical objects and properties.

1. Origin of coordinates:

Point O(0,0,0).

In a form of equations, it's a system of three equations

x = 0

y = 0

z = 0

2. Equation that describes all points on XY-plane:

For any point on XY-plane a plane perpendicular to Z-axis is that same XY-plane.

It intersects Z-axis at an origin of coordinates, so its Z-coordinate must be equal to zero.

So, the equation of XY-plane is

z = 0

3. Equation of an angle bisector between Y-axis and Z-axis:

Since all points within YZ-plane are characterized by an equation x = 0 and within YZ-plane points lying on an angle bisector between Y-axis and Z-axis are characterized by a property y = z, the final numerical representation of this line is a system of two linear equations

x = 0

y = z

4. Equation describing points on a sphere of a radius R with a center at the origin of coordinates:

The square of a distance from the origin of coordinates to a point A with coordinates (x,y,z) is, as follows from Pythagorean Theorem applied twice, equals to x²+y²+z². Therefore, an equation for a sphere of radius R is

x²+y²+z² = R²

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