Affine Geometry

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# Affine Geometry

Wai-Shing Luk

2018-09-12 📅

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Geometry
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-   Algebraic geometry (curve)
-   Incidence geometry (+ circle)
-   Projective geometry (+ straight)
    -   Notion of “Cross ratio”
-   Affine Geometry (+ parallel)
    -   Notion of “ratio”, “barycenter”, “convex”
-   Metric Geometry (+ orthogonal)

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Affine Space
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-   An affine space is either the degenerate space reduced to the empty
    set, or a triple $\langle P, V, + \rangle$
-   consisting of a nonempty set $P$ (points), a vector space $V$ (of
    translations or free vectors), and an action
    $+: P \times V \rightarrow P$, satisfying the following conditions:
  1.  $a + 0 = a$, for every $a \in P$
  2.  $(a + u) + v = a + (u + v)$, for every $a \in P$, $u, v \in V$
  3.  For any 2 points $a, b \in P$, there is a unique $u \in V$ such that
    $b = a + u$ (so we can define a minus operator such that $b - a$
    returns $u$.)

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Affine Space in C++ Concept
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```cpp
template <class P, class V, class F = typename V::value_type>
concept bool Affine_Space()
{
  return Vector_Space<V,F>()
      && requires (P p, V v) { { p + v } → P; }
      && requires (P p, P q) { { p - q } → V; };
}

template <class P, class V>
  requires Affine_Space<P,V>()
void Affine_Space_test(P p, V u, V v)
{
    assert( p + V(0) == p );
    assert( (p + u) + v == p + (u + v) );
}
```

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📚 Example 1: Affine Space in $R^3$
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-   A point $a = (a_1, a_2, a_3)$
-   A vector $v = [v_1, v_2, v_3]$
-   Addition: $a + v = (a_1 + v_1, a_2 + v_2, a_3 + v_3)$
    -   $a + 0 = a$
    -   $(a + u) + v = a + (u + v)$
    -   $b = a + (a \rightarrow b)$

> **Note**: Don’t access the coordinates as possible.

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📚 Example 2: An Unusual Example
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-   A set $P$ consisting of all points $(x, y, z)$ satisfying the
    equation $x^2 + y^2 - z = 0$
-   Define the action **+** on $P$ such that
    -   $(x, y, x^2+y^2)$ **+** $[u, v]$ =
        $(x + u, y + v, (x + u)^2 + (y + v)^2)$

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📚 Example 3: C/C++ Pointer Different
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-   Probably the most simplest affine space!
    -   `<T*> = <T*> + <int>`
    -   `<int> = <T*> – <T*>`
-   Avoid pointer arithmetic
    -   “Mid-point” of two pointers: `p + (p – q)/2` instead of
        `(p + q)/2`
-   Similarly, avoid direct coordinate arithmetic as possible! (How?)

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How to Avoid Coordinate Arithmetic?
-----------------------------------

-   Sol’n: Don’t provide arithmetic operators for `<Point>` class,
    except `operator-()`, which returns a `<vector>`
-   Advantages:
    -   Reduce the chance of overflow for integer coordinates (e.g.
        mid-point = `p + (p – q)/2` )
    -   More economic for calculation (e.g. triangle area need 2
        multiplications instead of 6)
    -   Reduce duplicate coding in `<Point>` and `<vector>`
    -   Follow concept in mathematics.

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# Q&A 🎤