Third projection of a point and a segment, competing points, visibility
Enter point coordinates: the calculator builds an epure with three projections. Or give two projections and it finds the third. If points coincide in one projection it shows which of them is visible.
Monge's method
In descriptive geometry an object is projected onto two or three mutually perpendicular planes: horizontal π₁, frontal π₂ and profile π₃. The planes are then folded into the sheet. The position of a point is given by three coordinates: x is the distance to the profile plane, y to the frontal one, z to the horizontal one.
What the third projection is
Two projections of a point already fix its position in space. They lie on one projection line perpendicular to the axis: A₁ and A₂ on a vertical one, A₂ and A₃ on a horizontal one. To find A₃, draw a horizontal line from A₂ and lay off the distance y from the z axis; on the epure the 45° line carries it over.
Octants
The three planes divide space into eight octants. In the first x, y, z are positive (the point is in front, above, left). The signs of the coordinates tell the octant at once, and so which projections are visible.
Competing points and visibility
Two points whose projections coincide in one plane are called competing. The one nearer to the viewer is visible: on π₁ the one with greater z, on π₂ with greater y, on π₃ with greater x. The hidden projection is written in brackets on the epure.
FAQ
How do I build the third projection of a point?
Knowing A₁ and A₂, draw a horizontal projection line from A₂. Lay off on it from the z axis the distance y (taken from A₁): that is A₃. On the epure the distance is carried over through the 45° line.
What are competing points?
Points whose projections coincide in one of the planes: two of the three coordinates are equal. They are used to decide which line or point is visible and which is hidden.
How do I know which of the competing points is visible?
On the horizontal projection the point with the greater height z is visible, on the frontal one the point with the greater depth y (nearer to the viewer), on the profile one the point with the greater x (further left).
Which octant does a point lie in?
By the signs of the coordinates: all plus is the first octant, in the second y is negative, in the third y and z are negative, in the fourth z is negative; with a negative x it is the fifth, sixth, seventh and eighth in the same order.
Why does the x axis go to the left on the epure?
It is the convention of Russian textbooks: the positive x axis goes left, y towards the viewer, z up. The calculator draws the epure that way.