do i also get enough precision to correctly "round the cube" using vector normalization? or do i have to figure out a method to "emulate" the correct curvature locally for precision?
Assessing Float32 Precision for Cube-to-Sphere Normalization on Earth Scale
The user is asking about normalizing vectors to round a cube into a sphere for an Earth-sized planet, specifically whether 32-bit float precision is sufficient for this operation when working with 32m quad-tree cells.
This is a critical precision question. Let me think through the math:
- We have a cube with Earth-sized dimensions (~12.7 million meters)
- We're subdividing down to 32m cells using quad-trees
- The user wants to normalize vectors to project cube faces onto a sphere
- They're concerned about precision loss during normalization
The key issue: When you normalize a vector from the center of the Earth to a point on the cube surface, you're dividing by a very large magnitude (~6.37 million meters). This amplifies floating-point errors.
Let me calculate the precision at the surface:
- At Earth's radius (6,371,000m), float32 ULP is about 0.5 meters
- This means position errors of up to ~0.5 meters when normalizing
But the real problem is vertex positioning. When you normalize the corners of a 32m cell:
- Corner position might be (x, y, z) where each component is ~6,371,000m
- Normalization involves: normalized = (x/r, y/r, z/r) where r ≈ 6,371,000m
- The result should lie on the unit sphere, but errors occur
For a 32m cell at the surface:
- The angular size is tiny: 32m / 6,371,000m ≈ 5.02e-6 radians ≈ 0.000288 degrees
- The resulting sphere-normalized vertices will be extremely close together
Precision analysis:
- When you normalize a vector like (6,371,000, 0, 0), you get (1.0, 0, 0) exactly in theory
- But for off-axis points like (6,371,000, 16, 16), the normalization involves dividing large numbers
- The relative error in float32 normalization is typically around 1-2 ULP
- For positions at Earth's radius, this translates to ~0.5-1 meter error in the final position
The critical insight: 32-bit float normalization is NOT precise enough for direct vertex positioning at 32m resolution. The error magnitude (~0.5-1m) is significant compared to your target resolution.
Better approaches:
- Local curvature approximation: Use the cube coordinates to compute spherical positions using double-precision on the CPU, then convert to float32 relative to local origin
- Geodetic calculations: Convert cube coordinates to latitude/longitude/height, then back to Cartesian using double precision
- Analytical sphere mapping: Use methods like "cube to sphere mapping" that preserve precision better
I should explain:
- Why direct normalization fails (quantify the error)
- What the error looks like in practice (vertex jitter, cracks)
- Better methods to handle it
- The fact that you need to compute vertex positions in double precision or use local approximations
The memory insertion is about the user's technical requirements for precision in planetary rendering. This is important context for future conversations about large-scale terrain rendering or procedural planet generation.
The user is asking about normalizing vectors to round a cube into a sphere for an Earth-sized planet, specifically whether 32-bit float precision is sufficient for this operation when working with 32m quad-tree cells.
This is a critical precision question. Let me think through the math:
- We have a cube with Earth-sized dimensions (~12.7 million meters)
- We're subdividing down to 32m cells using quad-trees
- The user wants to normalize vectors to project cube faces onto a sphere
- They're concerned about precision loss during normalization
The key issue: When you normalize a vector from the center of the Earth to a point on the cube surface, you're dividing by a very large magnitude (~6.37 million meters). This amplifies floating-point errors.
Let me calculate the precision at the surface:
- At Earth's radius (6,371,000m), float32 ULP is about 0.5 meters
- This means position errors of up to ~0.5 meters when normalizing
But the real problem is vertex positioning. When you normalize the corners of a 32m cell:
- Corner position might be (x, y, z) where each component is ~6,371,000m
- Normalization involves: normalized = (x/r, y/r, z/r) where r ≈ 6,371,000m
- The result should lie on the unit sphere, but errors occur
For a 32m cell at the surface:
- The angular size is tiny: 32m / 6,371,000m ≈ 5.02e-6 radians ≈ 0.000288 degrees
- The resulting sphere-normalized vertices will be extremely close together
Precision analysis:
- When you normalize a vector like (6,371,000, 0, 0), you get (1.0, 0, 0) exactly in theory
- But for off-axis points like (6,371,000, 16, 16), the normalization involves dividing large numbers
- The relative error in float32 normalization is typically around 1-2 ULP
- For positions at Earth's radius, this translates to ~0.5-1 meter error in the final position
The critical insight: 32-bit float normalization is NOT precise enough for direct vertex positioning at 32m resolution. The error magnitude (~0.5-1m) is significant compared to your target resolution.
Better approaches:
- Local curvature approximation: Use the cube coordinates to compute spherical positions using double-precision on the CPU, then convert to float32 relative to local origin
- Geodetic calculations: Convert cube coordinates to latitude/longitude/height, then back to Cartesian using double precision
- Analytical sphere mapping: Use methods like "cube to sphere mapping" that preserve precision better
I should explain:
- Why direct normalization fails (quantify the error)
- What the error looks like in practice (vertex jitter, cracks)
- Better methods to handle it
- The fact that you need to compute vertex positions in double precision or use local approximations
The memory insertion is about the user's technical requirements for precision in planetary rendering. This is important context for future conversations about large-scale terrain rendering or procedural planet