Hence, thought of as a line in a particular plane **yp, any tangent to Q has a unique image and moreover this image is the same for all planes through L.

Clearly, any line, l, of any bundle having one of these points of tangency, T, as vertex will be transformed into the entire pencil having the image of the second intersection of L and Q as vertex and lying in the plane determined by the image point and the generator of Af which is tangent to **zg at T.

If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).

For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is defined to be the smallest such distance.

For example, if the given manifold is a 2-sphere, one can picture the tangent space at a point as the plane which touches the sphere at that point and is perpendicular to the sphere's radius through the point.

In geometry, the tangent line ( or simply the tangent ) to a plane curve at a given point is the straight line that " just touches " the curve at that point — that is, coincides with the curve at that point and, near that point, is closer to the curve that any other line passing through that point.

The tangent would, of course, be correct if the principal planes were really plane.

There exists a circle in the osculating plane tangent to γ ( s ) whose Taylor series to second order at the point of contact agrees with that of γ ( s ).

Any non-singular curve on a smooth surface will have its tangent vector T lying in the tangent plane of the surface orthogonal

The normal curvature, k < sub > n </ sub >, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u ; the geodesic curvature, k < sub > g </ sub >, is the curvature of the curve projected onto the

surface's tangent plane ; and the geodesic torsion ( or relative torsion ), τ < sub > r </ sub >, measures the rate of change of the surface normal around the curve's tangent.

All curves with the same tangent vector will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing T and u.

This is a quadratic form in the tangent plane to the surface at a point whose value at a particular tangent vector X to the surface is the normal component of the acceleration of a curve along the surface tangent to X ; that is, it is the normal curvature to a curve tangent to X ( see above ).

A related notion of curvature is the shape operator, which is a linear operator from the tangent plane to itself.

The principal curvatures are the eigenvalues of the shape operator, and in fact the shape operator and second fundamental form have the same matrix representation with respect to a pair of orthonormal vectors of the tangent plane.

For two-and three-dimensional collisions the velocities in these formulas are the components perpendicular to the tangent line / plane at the point of contact.

Indeed, the equation means that the acceleration of the curve has no components in the direction of the surface ( and therefore it is perpendicular to the tangent plane of the surface at each point of the curve ).

* Curves or functions in the plane are orthogonal at an intersection if their tangent lines are perpendicular at that point.

Some authors define stereographic projection from the north pole ( 0, 0, 1 ) onto the plane z = − 1, which is tangent to the unit sphere at the south pole ( 0, 0, − 1 ).

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

Further, we see by Lemma 2 that the multiplicity of F can only change at a tangent point, and at such a point can only change by an even integer.

It is interesting that a 1: 1 correspondence can be established between the lines of two such pencils, so that in a sense a unique image can actually be assigned to each tangent.

Now the only way in which all curves of the image family of Af can pass through a fixed point is to have a generator of Af which is not a secant but a tangent of **zg, for then any point on such a generator will be transformed into the point of tangency.

Equivalently, we can think of tangent vectors as tangents to curves, and write

They can be thought of as alternating, multilinear maps on k tangent vectors.

The volume of the note can be changed by striking harder or softer, and the pitch can also be affected by varying the force of the tangent against the string ( known as Bebung ).

Since the string vibrates from the bridge only as far as the tangent, multiple keys with multiple tangents can be assigned to the same string.

Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot.

Any convex polyhedron can be distorted into a canonical form, in which a midsphere ( or intersphere ) exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.

In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit ( the stable manifold ) and another of the points that diverge from the orbit ( the unstable manifold ).

the double angle formula for the tangent can be written as

It can be defined because Lie groups are manifolds, so have tangent spaces at each point.

To every Lie group, we can associate a Lie algebra, whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group.

# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.

This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as − 1 on the tangent space T < sub > e </ sub >.

Using differential calculus, we can determine the limit, or the value that Δy / Δx approaches as Δy and Δx get closer to zero ; it follows that this limit is the exact slope of the tangent.

A vector in this tangent space can represent a possible velocity at x.

In differential geometry, one can attach to every point x of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible " directions " at which one can tangentially pass through x.

Any other point of intersection between C and Af will be called a tangent point.

Now Af and Af must both be tangent points on the T component in the f-plane ; ;

Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.

The tension at c is tangent to the curve at c and is therefore horizontal, and it pulls the section to the left so it may be written (− T < sub > 0 </ sub >, 0 ) where T < sub > 0 </ sub > is the magnitude of the force.

Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly using this method.

The most common geometric arrangement is where some convex polyhedron is in its canonical form, which is to say that the all its edges must be tangent to a certain sphere whose centre coincides with the centre of gravity ( average position ) of the tangent points.