## Monday, May 4, 2015

### Unizor - Geometry3D - Elements - Construction in 3D

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Construction in three-dimensional space is a more complicated task than on a plane. Two-dimensional construction on a plane can be modeled using paper, straight ruler and compass. In three dimensions this is not that easy. Lots of things we just have to imagine.

Let's start with a simple principle. If we have to construct something on a plane that already exists in three-dimensional space, we will just use the same techniques of construction as we used before when studying the geometry on a plane.

Next is to be able to construct a plane in three-dimensional space. Here we will use axioms and simple theorems we introduced in the beginning and just say that a plane should be considered constructed if, at least, one of these conditions is met:
(a) there are three points that are known to belong to a plane;
(b) there is a line and a point outside of this line that are known to belong to a plane;
(c) there are two intersecting lines that are known to belong to a plane;
(d) there are two parallel lines that are known to belong to a plane.

Each of the above conditions is sufficient to define one and only one plane in three-dimensional space, and that's why we consider our job of constructing a plane completed if one of them is satisfied. So, when asked to construct a plane, we can always resort to one of the above cases and consider the plane is constructed if one of those conditions is met.

But there are other three-dimensional figures we might need to construct. Let's examine the most common ones.
(1) A right prism can be considered constructed if we can construct a polygon in its base and know its altitude (height).
(2) A right circular cylinder can be considered constructed if we can construct a circle in its base and know its altitude (height).
(3) A right regular pyramid can be considered constructed if we can construct a polygon in its base and know its altitude (height).
(4) A right circular cone can be considered constructed if we can construct a circle in its base and know its altitude (height).
(5) A sphere can be considered constructed if we know the position of its center and its radius.

Just as an example, if a problem states that we have to construct a sphere tangential to all faces of a given regular tetrahedron from inside, we have to find a center and a radius of this sphere - and the problem is considered solved.