3D constructions

On a plane, we use to construct with compass and straightedge. This unfortunately restricts the constructible points : it is impossible to trisect angle in general, to square a circle or to double a cube with just these tools.
In 3D space, there is no more a sheet of paper to draw on it, and any construction is imaginary. We shall here generalize geometric constructions to 3D space.

A point on the plane corresponds to a line in 3D space. We'll define a "3D compass" as a drawing line, turning around an axis. This allows to "draw" generally an hyperboloïd. It may become a cone if the drawing line meets the axis, or into a plane if also perpendicular to axis. If the line is parallel to axis, it draws a cylinder. We also draw their intersections : lines, circles, points etc...
The openings of our 3D compass are not measured, so that we need 4 known points to use it : 2 to define axis and 2 to define the drawing line. Specifically, we can not adjust from scratch the "line parallel to axis" or "line perpendicular to axis" conditions. However we are allowed to move along the already adjusted device, in order to draw the same hyperboloïd somewhere else.

Using this device, construct the plane through three given points ABC     Hint

Top level
Double the cube, that is construct a segment with length AB × 3√2     Hint



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