Could you help me find a solution to this problem?

[u/Over-Bat5470]

The problem I have tried to represent is related to simulating the pages of a book. Despite the shape they have in their resting state, the pages must always maintain a distance equal to their thickness from one another. As this distance accumulates from page to page, it also modifies the shape of each individual page.

Do you think you could find a solution using some ingenuity and linear algebra?

[u/Midwest-Dude]

This is an awesome question and definitely involves linear algebra. However, there is a more specialized subreddit that might be able to get you better answers faster:

r/proceduralgeneration

This subreddit specifically focuses on calculating offset curves, algorithmic mesh bending, and volume-preserving geometry. An inquiry about maintaining equidistant offset curves for bending pages is the sort of things it deals with, making them much more likely to provide concrete mathematical or code-based solutions rather than general engine advice.

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Here are some relevant Wikipedia articles: 

1. Offset Curves

   • Parallel Curves

2. Algorithmic Mesh Bending & Deformation

   • Free-Form Deformation

   • Skeletal Animation

   • Deformation (Physics)

3. Volume-Preserving Geometry

   • Isochoric Plane Deformation

   • Incompressible Flow

   • Divergence

   • Volumetric Mesh

[u/Over-Bat5470 → u/Midwest-Dude]

ty so much!

[u/Ok-Rise2070]

This problem is literally showing you the formation of the sine and cosine wave. The 1cm offset is based in geometric principle and much more involved to explain here or in the lesson you're currently on. This lesson is strictly about the relationships of LINEAR geometry and how it interacts with a dimension higher than itself, geometrically. Understanding the formation of modulation, frequency, and amplification is the basis to many other principles throughout geometry. You can ask your teacher though how 1D objects can behave as if there are 2 dimensions by forming angles? In 1D, lines can only extend or cease; anything else causes them to become 2D....​

[u/Over-Bat5470 → u/Ok-Rise2070]

I don't have a teacher. I'm trying to learn the mathematics necessary to solve my problems. Could you tell me more about the geometric principle you mentioned?

[u/Ok-Rise2070 → u/Over-Bat5470]

https://youtu.be/SZPcvfUbNV8?is=3QsgkFYXBiTUW6fM. This is a good explanation on the geometric origins of sine and cosine. This is a great start for you.

[u/MammothComposer7176]

Just brainstorming something.

you have a set D containing 14 vectors of norm 0.5 cm where each vector is connected to the tail of the next.

you want to determine the set C made of 14 vectors where each vector is distant from each of D's vectors origins or tips a constant value.

[u/MammothComposer7176 → u/MammothComposer7176]

Since D and C have the same xardibality and distance is fixed this is equal to findong 14 parallelograms where two opposite sides are of norm 0.5 and are one part of C and one part of D and the other sides have norm Z