PhyslibAlpha.SpaceAndTime.Space.Surfaces.SphericalCylinder

The map $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \text{Space } 3$ embeds a unit circular cylinder into 3-dimensional Euclidean space. For a pair $(p, z)$, where $p$ is a point on the unit circle $S^1 \subset \text{Space } 2$ and $z \in \mathbb{R}$ is a real number representing the vertical coordinate, the function maps to a point in $\text{Space } 3$ by placing the coordinates of $p$ into the first two slots and $z$ into the third coordinate slot (index 2).

The embedding of the spherical cylinder $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \text{Space } 3$ is equal to the composition of the map $(p, z) \mapsto (z, \iota(p))$ with the inverse of the coordinate slicing map $\text{slice}(2)$. Specifically, for a point $p$ on the unit circle $S^1 \subset \text{Space } 2$ and a vertical coordinate $z \in \mathbb{R}$, the map is given by: $$\text{sphericalCylinder}(p, z) = \text{slice}(2)^{-1}(z, \text{sphericalShell}(2, p))$$ where $\text{sphericalShell}(2, p)$ is the canonical inclusion of $p$ into $\text{Space } 2$, and $\text{slice}(2)^{-1} : \mathbb{R} \times \text{Space } 2 \to \text{Space } 3$ is the isomorphism that places the first component of the product into the third coordinate slot (index 2) and the second component into the remaining slots.

The map $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \text{Space } 3$, which embeds the unit circular cylinder into 3-dimensional Euclidean space by mapping a point $p$ on the unit circle $S^1 \subset \text{Space } 2$ and a height $z \in \mathbb{R}$ to a point in $\text{Space } 3$, is injective.

The map $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \text{Space } 3$, which embeds the unit circular cylinder into 3-dimensional Euclidean space by mapping a point $p$ on the unit circle $S^1 \subset \text{Space } 2$ and a vertical coordinate $z \in \mathbb{R}$ to a point in $\text{Space } 3$, is continuous.

The map $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \text{Space } 3$ is a measurable embedding, where $S^1 = \{p \in \text{Space } 2 \mid \|p\| = 1\}$ denotes the unit circle in 2-dimensional Euclidean space. This implies that the map is injective and that a set $A \subseteq S^1 \times \mathbb{R}$ is measurable if and only if its image under the map is measurable in $\text{Space } 3$.

For any point $x = (p, z)$ in the product space $S^1 \times \mathbb{R}$, where $S^1 = \{ p \in \text{Space } 2 : \|p\| = 1 \}$ is the unit circle and $z \in \mathbb{R}$ is the vertical coordinate, the Euclidean norm of its image under the embedding $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \text{Space } 3$ is given by $$\|\text{sphericalCylinder}(x)\| = \sqrt{\|z\|^2 + 1}$$ where $\|z\|$ denotes the absolute value of the real component $x.2$.

The measure on $\text{Space } 3$ is defined as the pushforward of the product measure on $S^1 \times \mathbb{R}$ via the embedding $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \text{Space } 3$. The product measure is the product of the surface measure on the unit circle $S^1 \subset \text{Space } 2$ and the standard Lebesgue measure on $\mathbb{R}$. This measure represents the area measure used for integration over the surface of the infinite unit spherical cylinder in 3-dimensional Euclidean space.

The measure $\mu_{\text{cyl}}$ on $\mathbb{R}^3$ (denoted as `sphericalCylinderMeasure`), which represents the surface area measure of the unit spherical cylinder $S^1 \times \mathbb{R}$, has temperate growth. This means that the measure $\mu_{\text{cyl}}$ satisfies a polynomial growth condition, typically characterized by the existence of some $n \in \mathbb{N}$ such that $\int_{\mathbb{R}^3} (1 + \|x\|^2)^{-n} \, d\mu_{\text{cyl}}(x) < \infty$, allowing it to define a tempered distribution.

The surface measure on the spherical cylinder in $\text{Space } 3$ (the unit cylinder $x^2 + y^2 = 1$) is $s$-finite. This means that the measure can be expressed as a countable sum of finite measures.

The tempered distribution on $\mathbb{R}^3$ (denoted as $\text{Space } 3$) defined by the integration of Schwartz functions against the surface measure of the unit spherical cylinder. For any Schwartz function $f \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$, the distribution maps $f$ to the integral $\int_{\mathbb{R}^3} f(x) \, d\mu_{\text{cyl}}(x)$, where $\mu_{\text{cyl}}$ is the `sphericalCylinderMeasure` corresponding to the cylinder $x^2 + y^2 = 1$ in $\mathbb{R}^3$.

For any Schwartz function $f \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$, the value of the spherical cylinder distribution applied to $f$ is equal to the integral of $f$ over $\mathbb{R}^3$ with respect to the spherical cylinder surface measure $\mu_{\text{cyl}}$: $$\text{sphericalCylinderDist}(f) = \int_{\mathbb{R}^3} f(x) \, d\mu_{\text{cyl}}(x)$$ The measure $\mu_{\text{cyl}}$ (represented by `sphericalCylinderMeasure`) is the surface measure associated with the infinite unit cylinder $x^2 + y^2 = 1$ in three-dimensional Euclidean space.

For any Schwartz function $f \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$, the application of the spherical cylinder distribution $\text{sphericalCylinderDist}$ to $f$ is equal to the integral of $f$ composed with the cylinder embedding over the product of the unit circle $S^1$ and the real line $\mathbb{R}$: $$\text{sphericalCylinderDist}(f) = \int_{S^1 \times \mathbb{R}} f(\text{sphericalCylinder}(x)) \, d(\mu_{S^1} \otimes \lambda)(x)$$ where $\text{sphericalCylinder} : S^1 \times \mathbb{R} \to \mathbb{R}^3$ is the map embedding the unit circle and the vertical coordinate into 3D space, $\mu_{S^1}$ is the surface measure on the unit circle $S^1 \subset \mathbb{R}^2$, and $\lambda$ is the Lebesgue measure on $\mathbb{R}$.