21 Scene: v0.1.25 → v0.1.26 — Manifold Datatracker

 # 21 Scene
 ## 21.01 Scene and `Coordinate` type
-A scene is a vector space of $\mathbb{R}^3$. We fix the right-handed orientation and an orthonormal basis, labeled $\hat x$, $\hat y$, and $\hat z$. A point in the space can be expressed as a 3-tuple $(x, y, z) = x\,\hat x + y\,\hat y + z\,\hat z$, where $x$, $y$, and $z$ are real numbers.
+A scene is a vector space $\mathbb{R}^3$. We fix the right-handed orientation and an orthonormal basis, labeled $\hat x$, $\hat y$, and $\hat z$. A point in the space can be expressed as a 3-tuple $(x, y, z) = x\,\hat x + y\,\hat y + z\,\hat z$, where $x$, $y$, and $z$ are real numbers.
 A `Coordinate` type represents a point in the scene and is an array of three real numbers. It is represented as
 ```ts
 type Coordinate = [number, number, number]
 ```
-where the three numbers correspond to $x$, $y$, and $z$, in this order.
+where the three numbers correspond to $x$, $y$, and $z$, in that order.
-The ground plane is the subspace of the scene spanned by the basis $\hat x$ and $\hat y$, i.e., the set $\{(x, y, 0)\ |\ x \in \mathbb{R},\; y \in \mathbb{R}\}$. The up direction is defined as the positive direction of the basis vector $\hat z$.
+The ground plane is the subspace of the scene spanned by the basis $\hat x$ and $\hat y$, i.e., the set $\{(x, y, 0)\ |\ x \in \mathbb{R},\; y \in \mathbb{R}\}$. A coordinate projected onto the ground plane can therefore be specified by a 2-tuple $(x, y)$, referred to as the ground plane coordinate.
+On a plane in the scene, a point can be expressed as a 2-tuple $(u, v)$ with respect to a given orthonormal basis $\hat u$ and $\hat v$, similar to above. A `Coordinate2D` type represents a point in such a plane and is an array of two real numbers. It is represented as
+```ts
+type Coordinate2D = [number, number]
+```
+where the two numbers correspond to $u$ and $v$, in that order. One can therefore use `Coordinate2D` to express points in the ground plane using the basis $\hat u = \hat x$ and $\hat v = \hat y$.
+For a plane in the scene, the up direction is defined as $\hat n = \hat u \times \hat v$, where $\times$ is the usual cross product and $\hat u$ and $\hat v$ are the orthonormal basis vectors mentioned above. The up direction of the scene is the up direction of the ground plane, i.e., the positive direction of the basis vector $\hat z$.
 ## 21.02 Coordinate Transformation
 A coordinate transformation with uniform scale in the scene defined above transforms a vector $\vec v = (v_x, v_y, v_z)$ to $s\, R \, \vec v + \vec t$, where $R$ is a rotation operator, $s$ is a real number for uniform scale, and $\vec t$ is the translation of the origin. Such a transformation can therefore be encoded in three pieces of information: translation $\vec t$, rotation $R$, and scale $s$.
 ### 21.02.01 Rotation and `Euler` Type
 Rotation can be represented by the type `Euler`, specifying the Euler angles as a 3-tuple $(x, y, z)$ in radians. The rotations are applied in the order of the $x$-axis, then the $y$-axis, then the $z$-axis. It can be represented as
 ```ts
 type Euler = [number, number, number]
 ```
 where the three numbers correspond to rotation in radians around the $x$-axis, $y$-axis, and $z$-axis, in that order. For each rotation, a positive value means counterclockwise rotation. A zero rotation on all axes implies that the axes of the transformed coordinate align with those of the scene coordinate.
+Rotation on a plane can also be represented by an Euler angle as a single number. The rotation is applied around the up-axis of the plane. A `Euler2D` represents such data as
+```ts
+type Euler2D = number
+```
+where a positive `Euler2D` value means counterclockwise rotation.
 ### 21.02.02 Scale and `Scale` Type
 The type `Scale` represents the scaling of the transformed coordinate about its local origin and is a real number, where a value of 1 means no scaling. It can be represented as

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# 21 Scene

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# 21 Scene

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## 21.01 Scene and `Coordinate` type

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## 21.01 Scene and `Coordinate` type

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A scene is a vector space of $\mathbb{R}^3$. We fix the right-handed orientation and an orthonormal basis, labeled $\hat x$, $\hat y$, and $\hat z$. A point in the space can be expressed as a 3-tuple $(x, y, z) = x\,\hat x + y\,\hat y + z\,\hat z$, where $x$, $y$, and $z$ are real numbers.

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A scene is a vector space $\mathbb{R}^3$. We fix the right-handed orientation and an orthonormal basis, labeled $\hat x$, $\hat y$, and $\hat z$. A point in the space can be expressed as a 3-tuple $(x, y, z) = x\,\hat x + y\,\hat y + z\,\hat z$, where $x$, $y$, and $z$ are real numbers.

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A `Coordinate` type represents a point in the scene and is an array of three real numbers. It is represented as

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A `Coordinate` type represents a point in the scene and is an array of three real numbers. It is represented as

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```ts

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```ts

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type Coordinate = [number, number, number]

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type Coordinate = [number, number, number]

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```

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```

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where the three numbers correspond to $x$, $y$, and $z$, in this order.

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where the three numbers correspond to $x$, $y$, and $z$, in that order.

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The ground plane is the subspace of the scene spanned by the basis $\hat x$ and $\hat y$, i.e., the set $\{(x, y, 0)\ |\ x \in \mathbb{R},\; y \in \mathbb{R}\}$. The up direction is defined as the positive direction of the basis vector $\hat z$.

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The ground plane is the subspace of the scene spanned by the basis $\hat x$ and $\hat y$, i.e., the set $\{(x, y, 0)\ |\ x \in \mathbb{R},\; y \in \mathbb{R}\}$. A coordinate projected onto the ground plane can therefore be specified by a 2-tuple $(x, y)$, referred to as the ground plane coordinate.

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On a plane in the scene, a point can be expressed as a 2-tuple $(u, v)$ with respect to a given orthonormal basis $\hat u$ and $\hat v$, similar to above. A `Coordinate2D` type represents a point in such a plane and is an array of two real numbers. It is represented as

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```ts

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type Coordinate2D = [number, number]

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```

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where the two numbers correspond to $u$ and $v$, in that order. One can therefore use `Coordinate2D` to express points in the ground plane using the basis $\hat u = \hat x$ and $\hat v = \hat y$.

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For a plane in the scene, the up direction is defined as $\hat n = \hat u \times \hat v$, where $\times$ is the usual cross product and $\hat u$ and $\hat v$ are the orthonormal basis vectors mentioned above. The up direction of the scene is the up direction of the ground plane, i.e., the positive direction of the basis vector $\hat z$.

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## 21.02 Coordinate Transformation

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## 21.02 Coordinate Transformation

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A coordinate transformation with uniform scale in the scene defined above transforms a vector $\vec v = (v_x, v_y, v_z)$ to $s\, R \, \vec v + \vec t$, where $R$ is a rotation operator, $s$ is a real number for uniform scale, and $\vec t$ is the translation of the origin. Such a transformation can therefore be encoded in three pieces of information: translation $\vec t$, rotation $R$, and scale $s$.

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A coordinate transformation with uniform scale in the scene defined above transforms a vector $\vec v = (v_x, v_y, v_z)$ to $s\, R \, \vec v + \vec t$, where $R$ is a rotation operator, $s$ is a real number for uniform scale, and $\vec t$ is the translation of the origin. Such a transformation can therefore be encoded in three pieces of information: translation $\vec t$, rotation $R$, and scale $s$.

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### 21.02.01 Rotation and `Euler` Type

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### 21.02.01 Rotation and `Euler` Type

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Rotation can be represented by the type `Euler`, specifying the Euler angles as a 3-tuple $(x, y, z)$ in radians. The rotations are applied in the order of the $x$-axis, then the $y$-axis, then the $z$-axis. It can be represented as

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Rotation can be represented by the type `Euler`, specifying the Euler angles as a 3-tuple $(x, y, z)$ in radians. The rotations are applied in the order of the $x$-axis, then the $y$-axis, then the $z$-axis. It can be represented as

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```ts

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```ts

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type Euler = [number, number, number]

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type Euler = [number, number, number]

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```

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```

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where the three numbers correspond to rotation in radians around the $x$-axis, $y$-axis, and $z$-axis, in that order. For each rotation, a positive value means counterclockwise rotation. A zero rotation on all axes implies that the axes of the transformed coordinate align with those of the scene coordinate.

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where the three numbers correspond to rotation in radians around the $x$-axis, $y$-axis, and $z$-axis, in that order. For each rotation, a positive value means counterclockwise rotation. A zero rotation on all axes implies that the axes of the transformed coordinate align with those of the scene coordinate.

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Rotation on a plane can also be represented by an Euler angle as a single number. The rotation is applied around the up-axis of the plane. A `Euler2D` represents such data as

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```ts

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type Euler2D = number

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```

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where a positive `Euler2D` value means counterclockwise rotation.

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### 21.02.02 Scale and `Scale` Type

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### 21.02.02 Scale and `Scale` Type

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The type `Scale` represents the scaling of the transformed coordinate about its local origin and is a real number, where a value of 1 means no scaling. It can be represented as

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The type `Scale` represents the scaling of the transformed coordinate about its local origin and is a real number, where a value of 1 means no scaling. It can be represented as

21 Scenev0.1.25 → v0.1.26