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Wireframe Model in 3D
Polygon Model in 3D

One subfield of computer science is 2D computer graphics. Vector models can be very affective for encoding 2D geometric shapes. We begin our discussion of 2D computer graphics by focusing on wireframe models. A wireframe model is created by specifying a vertex matrix and edge table.

Wireframe Model in 2D

In this example, we will create a triangle using a wireframe model. We will use the vertexes from 3.1.1 Vertex Model of Points:

\begin{aligned} {\vec{v}}_{1} = [\begin{array}{c} 0 \\ 0 \end{array}] & {\vec{v}}_{2} = [\begin{array}{c} - 1 \\ 1 \end{array}] & {\vec{v}}_{3} = [\begin{array}{c} 2 \\ 1 \end{array}] \end{aligned}

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390479cd01ced13c64236fbaaed3aafa5a43245a: ${\vec{v}}_{1}$
b8b113db58287809b44243063d1974450d176baf: ${\vec{v}}_{2}$
5c41b34ba97ca8622399cfa815dfa811fde7040e: ${\vec{v}}_{3}$
0f8f8f5c6ed5e80c679f8e54649f72374c6d95f4: $u_{1}$
3b87b040aee9b2a4eb893d8ab53aba7a8b088bd9: $u_{2}$
50eb7876289650fcef6936fe6675261b606f1674: $u_{4}$
9cb8418b2d971489c553a80d798dbfc1ba984863: $u_{3}$
900307fc7db61df0da83813e9396c07fa243a877: $e_{1}$
9270a528e052d5f9f74169cea4670113cec6ed76: $e_{2}$
8e5065926355c57f048f75ddd7b58546c7e045c2: $e_{4}$
6ac6ccb77a0d5d361c9c1ab993422cefcc3901ed: $e_{5}$
fd71834e3854c793dccb0ff4b77a6a9884754adb: $e_{3}$
3e3e87b5a37796bb10bf21306efe03dda9b7fa2c: $e_{6}$
f2a0bfe9d38acd431768cf59caaa20c5c3b905db: $u_{j}$
2a9921dd768987dc5143ee2f47601f7022e653ff: $u_{k}$
663d27fc4e6238a96ee08cb5a9c0212e584648ea: $u_{1}$
f46bbfbed8baeb8a8909295fdb280f1d4653d3b5: $u_{2}$
583c3b266fbec0a3e557d1933329381cea56e145: $u_{4}$
58ac8859f8870a71ad16e2a7a038cba44e805d36: $u_{3}$
abb31f2b1eed1de1af16f7f31404fe5cf2126da1: $e_{1}$
924cf332c24ce8d1dca7dd5e9066e67ab56d224e: $e_{2}$
0faa5757f10897e78d1a958a11b23de2eb8785fb: $e_{3}$
64d079b1d130fee951ac766c786cec5998763d82: $e_{4}$
9d2b1f92d2ce2fc0bf88e8d93d8ad2e30079beee: $e_{5}$

This information can be encoded in a Vertex Table.

\begin{array}{ccc} Vertex 1 & Vertex 2 & Vertex 3 \\ 1st Coordinate & 0 & - 1 & 2 \\ 2nd Coordinate & 0 & 1 & 1 \end{array}

We can also indicate the edges (connections) between the vertices using an Edge Table.

\begin{array}{ccc} Edge & Start Vertex & End Vertex \\ 1 & 1 & 2 \\ 2 & 2 & 3 \\ 3 & 3 & 1 \end{array}

Remark. Notice that we specify the start and end vertices for each edge separately. In this case, we do not care about creating directed edges. Thus, the visual representation of the triangle contains lines connecting each vertex (thought of as "wires"). If we wanted to, we could use arrows that run from the starting vertex toward the end vertex, known as the directed edge. However, this is unnecessary for this model.
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Remark. Also, notice this model has no "area" since the shape contains only vertices and edges. This is why we call this a "frame" since it only specifies the outer region of the shape.
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From this model, we can generate a vertex matrix

V = [\begin{matrix} 0 & - 1 & 2 \\ 0 & 1 & 1 \end{matrix}]

We can geometrically transform this matrix using matrix operators, such as matrix-matrix multiplication or matrix-matrix addition. As long as the edge configurations don't change, we can keep the edge table untouched while executing any transformation!

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Remark. Also, notice this model has no "area" since the shape contains only vertices and edges. This is why we call this a "frame" since it only specifies the outer region of the shape. ⧫

Remark. Also, notice this model has no "area" since the shape contains only vertices and edges. This is why we call this a "frame" since it only specifies the outer region of the shape.
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