Parallel curve

• ${\displaystyle {\vec {x}}'_{d}(t)\parallel {\vec {x}}'(t),\quad }$ that means: the tangent vectors for a fixed parameter are parallel.
• ${\displaystyle k_{d}(t)={\frac {k(t)}{1+dk(t)}},\quad }$ with ${\displaystyle k(t)}$ the curvature of the given curve and ${\displaystyle k_{d}(t)}$ the curvature of the parallel curve for parameter ${\displaystyle t}$.
• ${\displaystyle R_{d}(t)=R(t)+d,\quad }$ with ${\displaystyle R(t)}$ the radius of curvature of the given curve and ${\displaystyle R_{d}(t)}$ the radius of curvature of the parallel curve for parameter ${\displaystyle t}$.
• When they exist, the osculating circles to parallel curves at corresponding points are concentric. [13]
• As for parallel lines, a normal line to a curve is also normal to its parallels.
• When parallel curves are constructed they will have cusps when the distance from the curve matches the radius of curvature. These are the points where the curve touches the evolute.
• If the progenitor curve is a boundary of a planar set and its parallel curve is without self-intersections, then the latter is the boundary of the Minkowski sum of the planar set and the disk of the given radius.

If the given curve is polynomial (meaning that ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$ are polynomials), then the parallel curves are usually not polynomial. In CAD area this is a drawback, because CAD systems use polynomials or rational curves. In order to get at least rational curves, the square root of the representation of the parallel curve has to be solvable. Such curves are called pythagorean hodograph curves and were investigated by R.T. Farouki.[14]

• ${\displaystyle |\operatorname {grad} h({\vec {x}})|=1\;,}$
• ${\displaystyle h({\vec {x}}+d\operatorname {grad} h({\vec {x}}))=h({\vec {x}})+d\;,}$
• ${\displaystyle \operatorname {grad} h({\vec {x}}+d\operatorname {grad} h({\vec {x}}))=\operatorname {grad} h({\vec {x}})\;.}$

Example:
The diagram shows parallel curves of the implicit curve with equation ${\displaystyle \;f(x,y)=x^{4}+y^{4}-1=0\;.}$
Remark: The curves ${\displaystyle \;f(x,y)=x^{4}+y^{4}-1=d\;}$ are not parallel curves, because ${\displaystyle \;|\operatorname {grad} f(x,y)|=1\;}$ is not true in the area of interest.

As described above, the parametric representation of a parallel curve, ${\displaystyle {\vec {x}}_{d}(t)}$, to a given curver, ${\displaystyle {\vec {x}}(t)}$, with distance ${\displaystyle |d|}$ is:

${\displaystyle {\vec {x}}_{d}(t)={\vec {x}}(t)+d{\vec {n}}(t)}$ with the unit normal ${\displaystyle {\vec {n}}(t)}$.

At a sharp corner (${\displaystyle t=t_{c}}$), the normal to ${\displaystyle {\vec {x}}(t_{c})}$ given by ${\displaystyle {\vec {n}}(t_{c})}$ is discontinuous, meaning the one-sided limit of the normal from the left ${\displaystyle {\vec {n}}(t_{c}^{-})}$ is unequal to the limit from the right ${\displaystyle {\vec {n}}(t_{c}^{+})}$. Mathematically,

${\displaystyle {\vec {n}}(t_{c}^{-})=\lim _{t\to t_{c}^{-}}{\vec {n}}(t)\neq {\vec {n}}(t_{c}^{+})=\lim _{t\to t_{c}^{+}}{\vec {n}}(t)}$.

Normal fan for defining parallel curves around a sharp corner

However, we can define a normal fan[11] ${\displaystyle {\vec {n}}_{f}(\alpha )}$ that provides an interpolant between ${\displaystyle {\vec {n}}(t_{c}^{-})}$ and ${\displaystyle {\vec {n}}(t_{c}^{+})}$, and use ${\displaystyle {\vec {n}}_{f}(\alpha )}$ in place of ${\displaystyle {\vec {n}}(t_{c})}$ at the sharp corner:

${\displaystyle {\vec {n}}_{f}(\alpha )={\frac {(1-\alpha ){\vec {n}}(t_{c}^{-})+\alpha {\vec {n}}(t_{c}^{+})}{\lVert (1-\alpha ){\vec {n}}(t_{c}^{-})+\alpha {\vec {n}}(t_{c}^{+})\rVert }},\quad }$where ${\displaystyle 0<\alpha <1}$.

The resulting definition of the parallel curve ${\displaystyle {\vec {x}}_{d}(t)}$ provides the desired behavior:

${\displaystyle {\vec {x}}_{d}(t)={\begin{cases}{\vec {x}}(t)+d{\vec {n}}(t),&{\text{if }}t<t_{c}{\text{ or }}t>t_{c}\\{\vec {x}}(t_{c})+d{\vec {n}}_{f}(\alpha ),&{\text{if }}t=t_{c}{\text{ where }}0<\alpha <1\end{cases}}}$

• ${\displaystyle {\partial {\vec {x}}_{d} \over \partial u}\parallel {\partial {\vec {x}} \over \partial u},\quad {\partial {\vec {x}}_{d} \over \partial v}\parallel {\partial {\vec {x}} \over \partial v},\quad }$ that means: the tangent vectors for fixed parameters are parallel.
• ${\displaystyle {\vec {n}}_{d}(u,v)=\pm {\vec {n}}(u,v),\quad }$ that means: the normal vectors for fixed parameters match direction.
• ${\displaystyle S_{d}=(1+dS)^{-1}S,\quad }$ where ${\displaystyle S_{d}}$ and ${\displaystyle S}$ are the shape operators for ${\displaystyle {\vec {x}}_{d}}$ and ${\displaystyle {\vec {x}}}$, respectively.

The principal curvatures are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gaussian curvature is its determinant, and the mean curvature is half its trace.

• ${\displaystyle S_{d}^{-1}=S^{-1}+dI,\quad }$ where ${\displaystyle S_{d}^{-1}}$ and ${\displaystyle S^{-1}}$ are the inverses of the shape operators for ${\displaystyle {\vec {x}}_{d}}$ and ${\displaystyle {\vec {x}}}$, respectively.

The principal radii of curvature are the eigenvalues of the inverse of the shape operator, the principal curvature directions are its eigenvectors, the reciprocal of the Gaussian curvature is its determinant, and the mean radius of curvature is half its trace.

Note the similarity to the geometric properties of parallel curves.

Assume you have a regular parametric representation of a curve, ${\displaystyle {\vec {x}}(t)=(x(t),y(t))}$, and you have a second curve that can be parameterized by its unit normal, ${\displaystyle {\vec {d}}({\vec {n}})}$, where the normal of ${\displaystyle {\vec {d}}({\vec {n}})={\vec {n}}}$ (this parameterization by normal exists for curves whose curvature is strictly positive or negative, and thus convex, smooth, and not straight). The parametric representation of the general offset curve of ${\displaystyle {\vec {x}}(t)}$ offset by ${\displaystyle {\vec {d}}({\vec {n}})}$ is:

${\displaystyle {\vec {x}}_{d}(t)={\vec {x}}(t)+{\vec {d}}({\vec {n}}(t)),\quad }$ where ${\displaystyle {\vec {n}}(t)}$ is the unit normal of ${\displaystyle {\vec {x}}(t)}$.

Note that the trival offset, ${\displaystyle {\vec {d}}({\vec {n}})=d{\vec {n}}}$, gives you ordinary parallel (aka, offset) curves.

• ${\displaystyle {\vec {x}}'_{d}(t)\parallel {\vec {x}}'(t),\quad }$ that means: the tangent vectors for a fixed parameter are parallel.
• As for parallel lines, a normal to a curve is also normal to its general offsets.
• ${\displaystyle k_{d}(t)={\dfrac {k(t)}{1+{\dfrac {k(t)}{k_{n}(t)}}}},\quad }$ with ${\displaystyle k_{d}(t)}$ the curvature of the general offset curve, ${\displaystyle k(t)}$ the curvature of ${\displaystyle {\vec {x}}(t)}$, and ${\displaystyle k_{n}(t)}$ the curvature of ${\displaystyle {\vec {d}}({\vec {n}}(t))}$ for parameter ${\displaystyle t}$.
• ${\displaystyle R_{d}(t)=R(t)+R_{n}(t),\quad }$ with ${\displaystyle R_{d}(t)}$ the radius of curvature of the general offset curve, ${\displaystyle R(t)}$ the radius of curvature of ${\displaystyle {\vec {x}}(t)}$, and ${\displaystyle R_{n}(t)}$ the radius of curvature of ${\displaystyle {\vec {d}}({\vec {n}}(t))}$ for parameter ${\displaystyle t}$.
• When general offset curves are constructed they will have cusps when the curvature of the curve matches curvature of the offset. These are the points where the curve touches the evolute.

General offset surfaces describe the shape of cuts made by a variety of cutting bits used by three-axis end mills in numerically controlled machining.[11] Assume you have a regular parametric representation of a surface, ${\displaystyle {\vec {x}}(u,v)=(x(u,v),y(u,v),z(u,v))}$, and you have a second surface that can be parameterized by its unit normal, ${\displaystyle {\vec {d}}({\vec {n}})}$, where the normal of ${\displaystyle {\vec {d}}({\vec {n}})={\vec {n}}}$ (this parameterization by normal exists for surfaces whose Gaussian curvature is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of ${\displaystyle {\vec {x}}(t)}$ offset by ${\displaystyle {\vec {d}}({\vec {n}})}$ is:

${\displaystyle {\vec {x}}_{d}(u,v)={\vec {x}}(u,v)+{\vec {d}}({\vec {n}}(u,v)),\quad }$ where ${\displaystyle {\vec {n}}(u,v)}$ is the unit normal of ${\displaystyle {\vec {x}}(u,v)}$.

Note that the trival offset, ${\displaystyle {\vec {d}}({\vec {n}})=d{\vec {n}}}$, gives you ordinary parallel (aka, offset) surfaces.

• As for parallel lines, the tangent plane of a surface is parallel to the tangent plane of its general offsets.
• As for parallel lines, a normal to a surface is also normal to its general offsets.
• ${\displaystyle S_{d}=(1+SS_{n}^{-1})^{-1}S,\quad }$ where ${\displaystyle S_{d},S,}$ and ${\displaystyle S_{n}}$ are the shape operators for ${\displaystyle {\vec {x}}_{d},{\vec {x}},}$ and ${\displaystyle {\vec {d}}({\vec {n}})}$, respectively.

The principal curvatures are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gaussian curvature is its determinant, and the mean curvature is half its trace.

• ${\displaystyle S_{d}^{-1}=S^{-1}+S_{n}^{-1},\quad }$ where ${\displaystyle S_{d}^{-1},S^{-1}}$ and ${\displaystyle S_{n}^{-1}}$ are the inverses of the shape operators for ${\displaystyle {\vec {x}}_{d},{\vec {x}},}$ and ${\displaystyle {\vec {d}}({\vec {n}})}$, respectively.

The principal radii of curvature are the eigenvalues of the inverse of the shape operator, the principal curvature directions are its eigenvectors, the reciprocal of the Gaussian curvature is its determinant, and the mean radius of curvature is half its trace.

Note the similarity to the geometric properties of general offset curves.

The geometric properties listed above for general offset curves and surfaces can be derived for offsets of arbitrary dimension. Assume you have a regular parametric representation of an n-dimensional surface, ${\displaystyle {\vec {x}}({\vec {u}})}$, where the dimension of ${\displaystyle {\vec {u}}}$ is n-1. Also assume you have a second n-dimensional surface that can be parameterized by its unit normal, ${\displaystyle {\vec {d}}({\vec {n}})}$, where the normal of ${\displaystyle {\vec {d}}({\vec {n}})={\vec {n}}}$ (this parameterization by normal exists for surfaces whose Gaussian curvature is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of ${\displaystyle {\vec {x}}({\vec {u}})}$ offset by ${\displaystyle {\vec {d}}({\vec {n}})}$ is:

${\displaystyle {\vec {x}}_{d}({\vec {u}})={\vec {x}}({\vec {u}})+{\vec {d}}({\vec {n}}({\vec {u}})),\quad }$ where ${\displaystyle {\vec {n}}({\vec {u}})}$ is the unit normal of ${\displaystyle {\vec {x}}({\vec {u}})}$. (The trival offset, ${\displaystyle {\vec {d}}({\vec {n}})=d{\vec {n}}}$, gives you ordinary parallel surfaces.)

First, notice that the normal of ${\displaystyle {\vec {x}}({\vec {u}})=}$ the normal of ${\displaystyle {\vec {d}}({\vec {n}}({\vec {u}}))={\vec {n}}({\vec {u}}),}$ by definition. Now, we'll apply the differential w.r.t. ${\displaystyle {\vec {u}}}$ to ${\displaystyle {\vec {x}}_{d}}$, which gives us its tangent vectors spanning its tangent plane.

${\displaystyle \partial {\vec {x}}_{d}({\vec {u}})=\partial {\vec {x}}({\vec {u}})+\partial {\vec {d}}({\vec {n}}({\vec {u}}))}$

Notice, the tangent vectors for ${\displaystyle {\vec {x}}_{d}}$ are the sum of tangent vectors for ${\displaystyle {\vec {x}}({\vec {u}})}$ and its offset ${\displaystyle {\vec {d}}({\vec {n}})}$, which share the same unit normal. Thus, the general offset surface shares the same tangent plane and normal with ${\displaystyle {\vec {x}}({\vec {u}})}$ and ${\displaystyle {\vec {d}}({\vec {n}}({\vec {u}}))}$. That aligns with the nature of envelopes.

We now consider the Weingarten equations for the shape operator, which can be written as ${\displaystyle \partial {\vec {n}}=-\partial {\vec {x}}S}$. If ${\displaystyle S}$ is invertable, ${\displaystyle \partial {\vec {x}}=-\partial {\vec {n}}S^{-1}}$. Recall that the principal curvatures of a surface are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gauss curvature is its determinant, and the mean curvature is half its trace. The inverse of the shape operator holds these same values for the radii of curvature.

Substituting into the equation for the differential of ${\displaystyle {\vec {x}}_{d}}$, we get:

${\displaystyle \partial {\vec {x}}_{d}=\partial {\vec {x}}-\partial {\vec {n}}S_{n}^{-1},\quad }$ where ${\displaystyle S_{n}}$ is the shape operator for ${\displaystyle {\vec {d}}({\vec {n}}({\vec {u}}))}$.

Next, we use the Weingarten equations again to replace ${\displaystyle \partial {\vec {n}}}$:

${\displaystyle \partial {\vec {x}}_{d}=\partial {\vec {x}}+\partial {\vec {x}}SS_{n}^{-1},\quad }$ where ${\displaystyle S}$ is the shape operator for ${\displaystyle {\vec {x}}({\vec {u}})}$.

Then, we solve for ${\displaystyle \partial {\vec {x}}}$ and multiple both sides by ${\displaystyle -S}$ to get back to the Weingarten equations, this time for ${\displaystyle \partial {\vec {x}}_{d}}$:

${\displaystyle \partial {\vec {x}}_{d}(I+SS_{n}^{-1})^{-1}=\partial {\vec {x}},}$

${\displaystyle -\partial {\vec {x}}_{d}(I+SS_{n}^{-1})^{-1}S=-\partial {\vec {x}}S=\partial {\vec {n}}.}$

Thus, ${\displaystyle S_{d}=(I+SS_{n}^{-1})^{-1}S}$, and inverting both sides gives us, ${\displaystyle S_{d}^{-1}=S^{-1}+S_{n}^{-1}}$.