Tangent Hyperplanes I: Geometry

Tangent Hyperplanes I: Geometry

tangent-space

Geometric foundations of tangent hyperplanes: configuration manifolds, vector fields, and infinitesimal state representations.

What a Tangent Space Describes

A tangent space describes the possible instantaneous velocities at a point on a smooth manifold. It is an exact derivative object. Using that space to predict a finite displacement is an approximation whose accuracy needs a separate bound. This distinction matters in a golf model: an admissible joint velocity at one configuration does not specify the finite swing path.

A hill gives a useful picture. A tangent plane matches the surface’s slope at the point of contact. A finite sheet of paper generally cannot fit a curved patch without a gap or deformation. Shrinking the patch makes the first-order mismatch vanish relative to its size; it does not make a finite patch perfectly flat.

Curves and Regular Constraints

For an embedded manifold $M\subset\mathbb R^n$, a tangent vector at $x$ is the velocity $\dot\gamma(0)$ of a differentiable curve satisfying $\gamma(0)=x$ and $\gamma(t)\in M$. Their collection is the vector space $T_xM$.

Suppose locally $M=\{x:h(x)=0\}$, where $h:\mathbb R^n\to\mathbb R^p$ is $C^1$, $p<n$, and its Jacobian $Dh(x)$ has full row rank $p$. The implicit function theorem gives a manifold near $x$ with dimension $n-p$. Differentiating $h(\gamma(t))=0$ gives

\[ Dh(x)\dot\gamma(0)=0, \qquad T_xM=\ker Dh(x). \]

Here $Dh$ is a $p\times n$ matrix. For multiple constraints this is matrix-vector multiplication, not a scalar dot product. Full rank is what makes the kernel precisely the tangent space of the level set.

The affine tangent space is $x+T_xM$. It passes through $x$ in the ambient space; it need not lie on $M$. It is a hyperplane only when its codimension is one. With two independent constraints in three dimensions, the affine tangent space is a line. The series title uses the familiar hyperplane picture, but the general construction is a tangent space.

Finite Displacements and Smoothness

Take $v\in T_xM$ with $\|v\|=\epsilon$. Under the regular $C^1$ assumptions, a local graph representation implies

\[ \operatorname{dist}(x+v,M)=o(\epsilon). \]

With $C^2$ regularity and bounded second derivatives in a sufficiently small regular neighborhood, there are constants $C>0$ and $\epsilon_0>0$ such that, for $0\le\epsilon\le\epsilon_0$,

\[ \operatorname{dist}(x+v,M)\le C\epsilon^2. \]

The constant depends on the geometry, coordinates and neighborhood. A bound on the constraint residual $\|h(x+v)\|$ also needs local conditioning of the constraint map before it becomes a distance bound. A small residual in arbitrarily rescaled equations is not itself a physical distance.

The quadratic statement is an upper bound. The leading coefficient may vanish in a particular direction, or the manifold may be flat. A factor-of-ten reduction in displacement therefore does not universally produce a factor-of-100 reduction in error.

Example: The Unit Circle

For $h(x_1,x_2)=x_1^2+x_2^2-1$, the derivative at $(1,0)$ is $[2\;0]$. Thus

\[ T_{(1,0)}S^1=\{(0,a):a\in\mathbb R\}. \]

The affine tangent line contains $P=(1,\epsilon)$. Its closest point on the circle is the radial projection

\[ y_* = \frac{(1,\epsilon)}{\sqrt{1+\epsilon^2}}, \]

and the Euclidean distance is

\[ \begin{aligned} \operatorname{dist}(P,S^1) &=\sqrt{1+\epsilon^2}-1\\ &=\frac{\epsilon^2}{2}-\frac{\epsilon^4}{8}+O(\epsilon^6). \end{aligned} \]

For example, $\epsilon=0.1$ gives approximately $0.00498756$ in the circle’s length units. The point $(\sqrt{1-\epsilon^2},\epsilon)$, when it exists, is the intersection at the same height, not the closest point. These two constructions agree only in their leading distance term near zero.

A Bad Constraint Description Can Hide Good Geometry

Loss of rank prevents this particular use of the implicit function theorem. It does not prove that the underlying set is singular. The equation $h(x,y)=y^3=0$ defines the smooth $x$-axis, although $Dh=0$ everywhere on it. Replacing it with $\tilde h(x,y)=y$ exposes the correct one-dimensional tangent space.

By contrast, intersecting branches or a cone vertex may fail to form a smooth manifold. In mechanics, distinguish a defective constraint representation from a genuine change in the set of admissible configurations before assigning a degree of freedom or interpreting a Jacobian null space.

From Golf Kinematics to Control

For a declared time-independent configuration constraint $h(q)=0$, admissible motion satisfies $Dh(q)\dot q=0$. A finite Euler step $q+\dot q\Delta t$ generally leaves a curved constraint surface; a constraint-preserving integration method must account for that geometry. A measured swing plane is not automatically an exact holonomic constraint, and configuration $q$ represents the modeled system rather than only the clubhead position.

Geometry alone does not guarantee a successful controller. Local stabilization through a linearized equilibrium model requires suitable stabilizability conditions. LQR additionally needs declared costs; applying its feedback to the nonlinear plant requires a region where the dynamics, constraints and actuator limits support the claimed behavior. Positive quadratic weights are design choices, not proof of physiological optimality.

Part 2 derives the first variation along a trajectory. Part 4 then separates its finite prediction error from measurement and model error.

Citation

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@online{untitled,
  author = {},
  title = {Tangent {Hyperplanes} {I:} {Geometry}},
  url = {https://affinedrift.com/articles/tangent-hyperplanes-series/part-1-geometry.html},
  langid = {en}
}

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--- title: "Tangent Hyperplanes I: Geometry" description: "Geometric foundations of tangent hyperplanes: configuration manifolds, vector fields, and infinitesimal state representations." aliases: - ../tangent-hyperplane-articles/Tangent_Hyperplanes.html - ../tangent-hyperplane-contraction/index.html - ../tangent-hyperplane-contraction/chapters/01-foundations.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Tangent_Hyperplanes_Series_Package/Landing_Manifesto.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Tangent_Hyperplanes_Series_Package/01_Tangent_Hyperplanes_I_Geometry.html categories: - tangent-space --- ## What a Tangent Space Describes A tangent space describes the possible instantaneous velocities at a point on a smooth manifold. It is an exact derivative object. Using that space to predict a finite displacement is an approximation whose accuracy needs a separate bound. This distinction matters in a golf model: an admissible joint velocity at one configuration does not specify the finite swing path. A hill gives a useful picture. A tangent plane matches the surface's slope at the point of contact. A finite sheet of paper generally cannot fit a curved patch without a gap or deformation. Shrinking the patch makes the first-order mismatch vanish relative to its size; it does not make a finite patch perfectly flat. ## Curves and Regular Constraints For an embedded manifold $M\subset\mathbb R^n$, a tangent vector at $x$ is the velocity $\dot\gamma(0)$ of a differentiable curve satisfying $\gamma(0)=x$ and $\gamma(t)\in M$. Their collection is the vector space $T_xM$. Suppose locally $M=\{x:h(x)=0\}$, where $h:\mathbb R^n\to\mathbb R^p$ is $C^1$, $p<n$, and its Jacobian $Dh(x)$ has full row rank $p$. The implicit function theorem gives a manifold near $x$ with dimension $n-p$. Differentiating $h(\gamma(t))=0$ gives $$ Dh(x)\dot\gamma(0)=0, \qquad T_xM=\ker Dh(x). $$ Here $Dh$ is a $p\times n$ matrix. For multiple constraints this is matrix-vector multiplication, not a scalar dot product. Full rank is what makes the kernel precisely the tangent space of the level set. The affine tangent space is $x+T_xM$. It passes through $x$ in the ambient space; it need not lie on $M$. It is a **hyperplane only when its codimension is one**. With two independent constraints in three dimensions, the affine tangent space is a line. The series title uses the familiar hyperplane picture, but the general construction is a tangent space. ## Finite Displacements and Smoothness Take $v\in T_xM$ with $\|v\|=\epsilon$. Under the regular $C^1$ assumptions, a local graph representation implies $$ \operatorname{dist}(x+v,M)=o(\epsilon). $$ With $C^2$ regularity and bounded second derivatives in a sufficiently small regular neighborhood, there are constants $C>0$ and $\epsilon_0>0$ such that, for $0\le\epsilon\le\epsilon_0$, $$ \operatorname{dist}(x+v,M)\le C\epsilon^2. $$ The constant depends on the geometry, coordinates and neighborhood. A bound on the constraint residual $\|h(x+v)\|$ also needs local conditioning of the constraint map before it becomes a distance bound. A small residual in arbitrarily rescaled equations is not itself a physical distance. The quadratic statement is an upper bound. The leading coefficient may vanish in a particular direction, or the manifold may be flat. A factor-of-ten reduction in displacement therefore does not universally produce a factor-of-100 reduction in error. ## Example: The Unit Circle For $h(x_1,x_2)=x_1^2+x_2^2-1$, the derivative at $(1,0)$ is $[2\;0]$. Thus $$ T_{(1,0)}S^1=\{(0,a):a\in\mathbb R\}. $$ The affine tangent line contains $P=(1,\epsilon)$. Its closest point on the circle is the radial projection $$ y_* = \frac{(1,\epsilon)}{\sqrt{1+\epsilon^2}}, $$ and the Euclidean distance is $$ \begin{aligned} \operatorname{dist}(P,S^1) &=\sqrt{1+\epsilon^2}-1\\ &=\frac{\epsilon^2}{2}-\frac{\epsilon^4}{8}+O(\epsilon^6). \end{aligned} $$ For example, $\epsilon=0.1$ gives approximately $0.00498756$ in the circle's length units. The point $(\sqrt{1-\epsilon^2},\epsilon)$, when it exists, is the intersection at the same height, not the closest point. These two constructions agree only in their leading distance term near zero. ## A Bad Constraint Description Can Hide Good Geometry Loss of rank prevents this particular use of the implicit function theorem. It does not prove that the underlying set is singular. The equation $h(x,y)=y^3=0$ defines the smooth $x$-axis, although $Dh=0$ everywhere on it. Replacing it with $\tilde h(x,y)=y$ exposes the correct one-dimensional tangent space. By contrast, intersecting branches or a cone vertex may fail to form a smooth manifold. In mechanics, distinguish a defective constraint representation from a genuine change in the set of admissible configurations before assigning a degree of freedom or interpreting a Jacobian null space. ## From Golf Kinematics to Control For a declared time-independent configuration constraint $h(q)=0$, admissible motion satisfies $Dh(q)\dot q=0$. A finite Euler step $q+\dot q\Delta t$ generally leaves a curved constraint surface; a constraint-preserving integration method must account for that geometry. A measured swing plane is not automatically an exact holonomic constraint, and configuration $q$ represents the modeled system rather than only the clubhead position. Geometry alone does not guarantee a successful controller. Local stabilization through a linearized equilibrium model requires suitable stabilizability conditions. LQR additionally needs declared costs; applying its feedback to the nonlinear plant requires a region where the dynamics, constraints and actuator limits support the claimed behavior. Positive quadratic weights are design choices, not proof of physiological optimality. [Part 2](./part-2-dynamics.html) derives the first variation along a trajectory. [Part 4](./part-4-residuals-curvature.html) then separates its finite prediction error from measurement and model error. ## Related Articles ::: {.callout-note title="See Also"} - **[Screw Theory Reference](../screw-theory-reference.html)** — Lie algebra se(3), twists, and infinitesimal generators of rigid-body motion - **[Tangent-Space Series](../../pages/tangent-hyperplanes.html)** — The series hub: full reading order, provenance, and companion material - **[The Null Space of the Constraint Jacobian](../null-space-constraint-jacobian.html)** — Constraint gradients and their kernels: the implicit-surface view of the same geometry - **[Degrees of Freedom and the Curse of Dimensionality](../degrees-of-freedom-and-dimensionality.html)** — How constraints and structure collapse effective dimensionality :::