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Intersections: line with plane and plane with plane

Two intersections come up in every exercise: line with plane and plane with plane. The first is solved by putting the line inside an auxiliary projecting plane; the second, with the points where the matching traces of both planes cross. The rest of the block is variations: planes parallel to the ground line, profile planes and planes that do not meet on the paper.

What you need to know

• Find the point where a line crosses a plane, using an auxiliary projecting plane.
• Draw the intersection line of two planes given by their traces.
• Solve the case where the traces are parallel and the cut falls outside the drawing.
• Check that the point found really belongs to both elements.

How it is solved

1. Line and plane: contain the line in a projecting plane (for example, a vertical one that shows it edge-on).
2. Find the intersection line of that auxiliary plane with the given one: where their matching traces cross.
3. That line and the original line meet at the point you are looking for. Check it is in both projections.
4. Plane with plane: join the crossing of the horizontal traces with that of the vertical ones; that is the intersection line.

How it appears in the exam

It is the basis of distances and angles: almost no exam exercise is solved without an intersection first. When the traces do not cross on the paper, extend the ground line or use an auxiliary plane; do not invent the point.

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