The affine lines y=3x+1 and y=3x−2 never meet in R². In homogeneous coordinates their equations are Y=3X+Z and Y=3X−2Z. Subtraction gives Z=0, and then Y=3X. They meet projectively at [1:3:0].
This is not an ordinary point very far away. It is a one-dimensional subspace of R³ representing a direction. Projective completion changes the point set and incidence relation in a precise algebraic way.
Predict
Predict whether [1:2:3] and [2:4:6] denote different points. They denote the same nonzero scalar class. Predict whether [0:0:0] is allowed. It is not: the zero vector spans no one-dimensional subspace.
Predict whether all affine parallel pairs acquire one common universal infinity point. Different directions give different points on the line Z=0.
Experiment
Move between an affine chart and homogeneous representatives.
Projective point
In chart Z≠0?
Affine interpretation
[2:4:2]
Yes
(1,2)
[−1:3:1]
Yes
(−1,3)
[1:3:0]
No
Direction slope three
[0:1:0]
No
Vertical direction
Normalize by dividing by Z only in the first two rows. For an ideal point, choose a different chart if coordinates are needed; division by zero is not a convention for infinity.
Schematic, not a metric plot: the two affine lines have no finite intersection. Their homogeneous equations share the ideal point [1:3:0], which is not drawn as a distant ordinary point.
Understand
Define RP² as the set of one-dimensional subspaces of R³. A projective line is the collection of one-dimensional subspaces inside a two-dimensional subspace. Equivalently it has an equation aX+bY+cZ=0 with a,b,c not all zero, considered up to nonzero scaling.
Distinct projective points have independent representatives p,q, and their unique joining line is the projectivization of span(p,q). Distinct projective lines correspond to distinct two-dimensional subspaces. The dimension formula shows their intersection has dimension one, hence exactly one projective point.
The chart Z≠0 identifies [X:Y:Z] with (X/Z,Y/Z). Its complement Z=0 is itself a projective line. An affine line P+tv acquires the ideal point [v₁:v₂:0]. Two affine lines of the same direction therefore meet there, while nonparallel affine lines retain their finite intersection.
Apply
An invertible matrix M acts on projective points by [p]↦[Mp]. The definition is independent of representative, and invertibility ensures the image is nonzero. It carries projective lines to projective lines.
Repair “Projective transformations preserve parallelism.” They preserve incidence, but may move the selected infinity line. A formerly ideal intersection can become finite in the original chart. Affine parallelism depends on the chart, not just on the projective plane.
For optional depth, not every abstract projective plane is coordinatized by a commutative field. Desarguesian and non-Desarguesian planes distinguish incidence axioms from the particular real model constructed here. We prove the real model’s properties, not a universal classification of projective planes.
Continue
Draw an affine chart with its ideal directions as a bookkeeping aid, then justify every incidence using subspaces or equations. Continue to duality and cross-ratio for projective statements that survive changing the chart.
Apply · explain · revise
Practice and reasoning
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Lines y=2x+1 and y=2x−4 meet projectively at [1:k:0]. Find k.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which triples represent the same projective point?
For independent nonzero vectors p,q in R³, justify that the projective line through [p],[q] consists of one-dimensional subspaces of span(p,q).
Structure your argument and preserve revisions
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Compute and justify the intersection of projective lines X+Y−Z=0 and X−Y=0.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show all affine lines with direction (a,b)≠(0,0) acquire the same ideal point in the chart Z≠0.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why a projective transformation may fail to preserve parallelism in a fixed affine chart.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
