1. Geometry INVARIANTS Parameter
Topology Continuity
Projective Geometry Spherical Geometry Straightedge 8
Affine Hyperbolic Elliptic
Geometry Geometry Geometry
Parallelism 6
Similarity Non-Euclidean
Geometry Geometry Angle 4
Congruence Distance 3
Geometry
Euclidean
Geometry

2. Congruence Geometry
INVARIANTS Para.
Transformation(Isometry)
Continuity
(Reflection, Rotation, Translation)
Straightedge
Parallelism
Angle
Distance 3

3. Congruence Geometry
INVARIANTS Para.
Transformation(Isometry) Continuity
(Reflection, Rotation, Translation)
Straightedge
Parallelism
Angle
Distance 3

4. DOF 3 Congruence Geometry INV 5
Parameters Visuals Invariants Examples Transformation
Continuity
3 Isometry
Iso+metry
Same+Measure
Straightness
1 rotation
Length Reflection
2 translation Parallelism
Area Rotation
𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 𝑡𝑥 Angle Angle Translation
𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑡𝑦
0 0 1
Distance
Glide Reflection
Euclidean geometry describes shapes “as they are”

5. Similarity Geometry INVARIANTS Para.
Continuity
Transformation Straightedge
Enlargement, Reduction
Parallelism
Angle 4
?

6. DOF 4 Similar Geometry INV 4
Parameters Visuals Invariants Examples Transformation
Continuity
4
ratios of
length
1 scale Straightness Isometry+Scale
angle
1 rotation
Parallelism ratios of Reduction
2 translation areas
parallel
Angle
lines Enlargement
?
It is the most general transformation that preserves shape..

7. Affine Geometry INVARIANTS Para.
Continuity
Transformation Straightedge
Shear, Stretch
Parallelism 6
?
?

8. DOF 6 Affine Geometry INV 3
Parameters Visuals Invariants Examples Transformation
Continuity
6
Ratio of area
Straightness Linear
2 scale Midpoint
Transforma
2 rotation Centroid tion
Parallelism
2 translation Line at infinity Affine
? parallel lines
Transforma
tion
ratios of parallel
? lengths

9. Projective Geometry
Transformation INVARIANTS
Continuity
Para.
Perspective, Projective Transformation
Straightedge 8
?
?
?

11. DOF 8 Projective Geometry INV 2
Parameters Visuals Invariants Examples Transformation
Continuity
8
Concurrency
Projective Transformation
2 scale Collinearity
Straightness Incidence
Perspectivity
Order of contact
2 rotation intersection,
?
tangency
2 translation inflection
? Cross ratio
2 line at conic section
infinity ?
Projective geometry describes objects “as they appear”
Line at infinity becomes finite, allows to observe vanishing points

12. Topology INVARIANTS
Continuity
Transformation ?
Topological Transformation
?
A ?
A B
?
C
B C