Interactive method
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1.1. Line parallel to a line (through a point P)
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Setup
Given a line r (projections r', r'') and an outside point P. We want to draw s || r through P.
1.2. Line parallel to a plane
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Key concept
A line is parallel to a plane if it is parallel to SOME line contained in that plane (not to all of them). To draw r through P parallel to Alpha, just draw an auxiliary line s in Alpha and make r || s.
1.3. Plane parallel to a plane
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Parallelism condition
Two planes are parallel if their matching traces are parallel (Alpha1 || Beta1 and Alpha2 || Beta2).
1.4. Plane parallel to a line
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Theory
A plane is parallel to a line if it contains a line parallel to the given one. If we are asked for a plane through P parallel to r, we look for an auxiliary line s // r through P. Any plane containing s (and not containing r) is parallel to r: a single line does not define the plane, you need a second direction.
⊥2. Perpendicularity
2.1. Line perpendicular to a plane
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Fundamental theorem
A line is perpendicular to a plane if its projections are perpendicular to the traces of the plane (r' ⊥ Alpha1, r'' ⊥ Alpha2). Directly.
2.4 Line perpendicular to another through a point
STEP 1 / 2- Option A: Horizontal or frontal line. If the line we want is parallel to a projection plane, the 90° angle is seen in true size in that plane.
- Option B (general): Perpendicular plane. We draw a plane perpendicular to “r” through “P”, and contain a line in it.
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General theory
To draw a line perpendicular to a given line (r) through a point (P), there are two main possibilities.
2.2. Plane perpendicular to a line
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Theory
A plane is perpendicular to a line if its traces are perpendicular to the matching projections of the line (Alpha1 ⊥ r' and Alpha2 ⊥ r''), through a point P.
2.3. Plane perpendicular to a plane
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Condition
A plane Beta is perpendicular to another plane Alpha if Beta contains a line perpendicular to Alpha. To draw it through P, just draw the line perpendicular to the given plane through P.
2.4. Line perpendicular to another through a point
STEP 1 / 3- Option A: Draw a horizontal or frontal line whose projections are perpendicular to the given one (theorem of the three perpendiculars).
- Option B (general): Draw a plane perpendicular to the given line through the point. Then contain any line in that plane through the point.
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Two possibilities
To draw a line perpendicular to a given one through a point there are 2 possibilities:
Summary of conditions
- Line ∥ plane: the line is parallel to some line of the plane.
- Plane ∥ plane: two intersecting lines of one are parallel to two of the other (or the matching traces are parallel).
- Line ⊥ plane: r′ ⊥ α₁ and r″ ⊥ α₂.
- Plane ⊥ line: α₁ ⊥ r′ and α₂ ⊥ r″.
- Plane ⊥ plane: one contains a line perpendicular to the other.
- Line ⊥ line: if one is parallel to a projection plane, the perpendicularity is seen in that view; if not, rotate it flat or use the perpendicular plane.