Question at position 6 Flattening a globe to two-dimensions involves compression.TrueFalseSingle choice
A
True
B
False

Question

Question at position 6 Flattening a globe to two-dimensions involves compression.TrueFalse

BlackTom AI · Accepted Answer

Gauss's Theorema Egregium shows a sphere (positive Gaussian curvature 1/R^2) cannot be mapped isometrically onto a plane (Gaussian curvature 0). That fundamental differential-geometry result is the core reason why flattening a globe to two dimensions must introduce distortion — compression or stretching — somewhere.

1) Relevant knowledge points and concepts
- Gaussian curvature: sphere K = 1/R^2, plane K = 0; an isometry must preserve curvature, so no isometric flattening exists.
- Map projections trade off preservation of properties: some preserve angles (conformal) but distort area, others preserve area but distort shape, and none preserve all metric properties.
- Practical intuition: peeling an orange rind to lie flat requires cutting and/or stretching — some regions compress or expand.

2) Step-by-step reasoning
- If you attempted to flatten the globe without compression/stretching, distances and areas in small neighborhoods would be preserved → local isometry.
- Gauss’s theorem forbids a local isometry between sphere and plane because curvature differs.
- Therefore any flattening must alter local metrics: some lengths/areas must be reduced (compression) or increased (stretching) relative to their spherical values.

3) Demonstration (intermediate steps)
- Sphere curvature K = 1/R^2; plane curvature K = 0.
- Because K_sphere ≠ K_plane, ∃ no mapping f that preserves the first fundamental form everywhere.
- Hence differential area elements dA_sphere cannot all equal corresponding dA_plane; at least some are compressed (dA_plane < dA_sphere) or stretched (dA_plane > dA_sphere).

4) Why the correct answer is correct
- The statement "Flattening a globe to two-dimensions involves compression" is true because any projection must change local metrics; compression is one form of that change and necessarily occurs in some parts of most projections (or globally for equal-area adjustments elsewhere).

5) Why the other option is incorrect
- "False" would claim a distortion-free flattening exists; that contradicts Gauss’s theorem and empirical behavior of map projections (e.g., Mercator hugely enlarges polar areas compared to the equator).

6) Conclusion
- The correct answer is True, matching the provided answer.

Question at position 6 Flattening a globe to two-dimensions involves compression.TrueFalseSingle choice

Similar Questions

Question at position 9 The Goode's homolosine map (pictured below or picture A on z.umn.edu/geog1502pics) is an example of what kind of projection? interrupted projection. planar projection.

Question at position 7 A map with an equivalent projection preserves the property ofarea.distance.scale.shape.

The following is what type of projection:

Equal-area projections do not show mapped areas in correct proportion to one another.

Conformal projections preserve shape and angle, but strongly distort are in a process.

Based on discussions in class, what kind of shape was the globe likely projected onto in this projection?

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