Constructing in space and GMT

Constructing in space refers to creating geometric objects (like points, lines, planes, spheres) and finding their positions or relationships in three-dimensional space. Problems often ask for the locus (set of all possible positions) of a point that satisfies certain conditions in space. GMT stands for Geometric Methods and Transformations, which are tools like symmetry, reflection, rotation, and projection used to solve spatial construction problems.

Important properties

• A locus in space can be a line, plane, sphere, or more complex surface, depending on the condition.

• Common constructions include: the set of points equidistant from two points (a plane), from a point and a plane (a sphere), or from two planes (another plane).

• Transformations (like reflection or rotation) can simplify spatial construction problems by reducing them to more familiar or symmetric cases.

• Projection (dropping perpendiculars or 'flattening' onto a plane) is often used to relate 3D problems to 2D geometry.