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tangent and point
Any other point of intersection between C and Af will be called a tangent point.
Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.
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.
In some neighborhood of an isolated tangent point in the f-plane, say Af, the function Af is either double-valued or has no values defined, except at the tangent point itself, where it is single-valued.
In the neighborhood of an end point of an interval of tangent points in the f-plane the function is two-valued or no-valued on one side, and is a single-valued function consisting entirely of tangent points on the other side.
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.
The most obvious of these is the quadratic complex of tangents to Q, each line of which is transformed into the entire pencil of lines tangent to Q at the image of the point of tangency of the given line.
Moreover, if Af and Af are two planes intersecting in a line l, tangent to Q at a point P, the two free intersections of the image curves Af and Af must coincide at P', the image of P, and at this point Af and Af must have a common tangent l'.
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.

tangent and Q
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.
Since two curves of symbol Af on Q intersect in Af points, it follows that there are Af lines of Af which are tangent to Aj.
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.
Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.
It can be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P, ( assuming that the first-derivative of the curve is continuous at point P so that there is only one tangent ).
As a consequence, one could say that the limit as Q approaches P of the secant's slope, or direction, is that of the tangent.
Already in this example whole sets have ill-defined mappings: taking a point P on Q as origin, we can use lines through P, intersecting Q at one other point, to project to a plane — but this definition breaks down with all lines tangent to Q at P, which in a certain sense ' blow up ' P into the intersection of the tangent plane with the plane to which we project.

tangent and occurs
Its absolute maximum size occurs at the moment expansion has stopped, and when graphed, its tangent slope is zero.
As the tonearm tracks the groove, the stylus exerts a frictional force tangent to the arc of the groove, and since this force does not intersect the tone arm pivot, a clockwise rotational force ( moment ) occurs and a reaction skating force is exerted on the stylus by the record groove wall away from center of the disc.
The speed for best range ( i. e., distance travelled ) occurs at the speed at which a tangent from the origin touches the fuel flow rate curve.
The least distortion occurs at the tangent point.

tangent and when
So the line is a horizontal tangent for the arctangent when x tends to −∞, and is a horizontal tangent for the arctangent when x tends to +∞.
Sometimes the vein will actually pass tangent to the notch, but often it will pass by at a small distance, and when that happens a spur vein ( occasionally a pair of such spur veins ) branches off and joins the leaf margin at the deepest point of the notch.
When applied to a tangent vector X to the surface, the shape operator is the tangential component of the rate of change of the normal vector when moved along a curve on the surface tangent to X.
When the axis is tangent to the circle, the resulting surface is called a horn torus ; when the axis is a chord of the circle, it is called a spindle torus.
In particular, when V is a unit vector, remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle.
Elongations occur when an inner planet ’ s position, in its orbital path, is at tangent to the view from Earth.
The contour lines of f and g touch when the tangent vectors of the contour lines are parallel.
We can also reparametrize geodesics to be unit speed, so equivalently we can define exp < sub > p </ sub >( v ) = β (| v |) where β is the unit-speed geodesic ( geodesic parameterized by arc length ) going in the direction of v. As we vary the tangent vector v we will get, when applying exp < sub > p </ sub >, different points on M which are within some distance from the base point p — this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of " linearization " of the manifold.
An important property of the exponential map is the following lemma of Gauss ( yet another Gauss's lemma ): given any tangent vector v in the domain of definition of exp < sub > p </ sub >, and another vector w based at the tip of v ( hence w is actually in the double-tangent space T < sub > v </ sub >( T < sub > p </ sub > M )) and orthogonal to v, remains orthogonal to v when pushed forward via the exponential map.
If R ( x ) is the remainder when P ( x ) is divided by ( x – r )< sup > 2 </ sup > — that is, by x < sup > 2 </ sup > – 2rx + r < sup > 2 </ sup > — then the equation of the tangent line to P ( x ) at x
Further, upper tangent arc cannot occur during nighttime when the observation was made.
In geometry, an inscribed angle is formed when two secant lines of a circle ( or, in a degenerate case, when one secant line and one tangent line of that circle ) intersect on the circle.
The general definition is that singular points of C are the cases when the tangent space has dimension 2.
In other words, we have a smooth tensor field J of rank ( 1, 1 ) such that J < sup > 2 </ sup > = − 1 when regarded as a vector bundle isomorphism J: TM → TM on the tangent bundle.
Therefore an even dimensional manifold always admits a ( 1, 1 ) rank tensor pointwise ( which is just a linear transformation on each tangent space ) such that J < sub > p </ sub >< sup > 2 </ sup > = − 1 at each point p. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined.

tangent and C
There are three possibilities: ( A ) Af remains tangent to C as it is translated ; ;
The first possibility results in a closed interval of tangent points in the f-plane, the end points of which fall into category ( B ) or ( C ).
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 ).
:# Draw the circumcircle ( tangent to every corner A, B, C and D ).
:# Draw lines tangent to the circumcircle at each corner A, B, C, D.
If M is an open subset of R < sup > n </ sup >, then M is a C < sup >∞</ sup > manifold in a natural manner ( take the charts to be the identity maps ), and the tangent spaces are all naturally identified with R < sup > n </ sup >.
where d is an infinitesimal vector line element of the closed curve C, with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C, with the sign determined by the integration direction.
The technical statement is as follows: if M is a given m-dimensional Riemannian manifold ( analytic or of class C < sup > k </ sup >, 3 ≤ k ≤ ∞), then there exists a number n ( with n ≤ m ( 3m + 11 )/ 2 if M is a compact manifold, or n ≤ m ( m + 1 )( 3m + 11 )/ 2 if M is a non-compact manifold ) and an injective map ƒ: M → R < sup > n </ sup > ( also analytic or of class C < sup > k </ sup >) such that for every point p of M, the derivative dƒ < sub > p </ sub > is a linear map from the tangent space T < sub > p </ sub > M to R < sup > n </ sup > which is compatible with the given inner product on T < sub > p </ sub > M and the standard dot product of R < sup > n </ sup > in the following sense:
The curvature of C at a point is a measure of how sensitive its tangent line is to moving the point to other nearby points.
Thus if γ ( s ) is the arclength parametrization of C then the unit tangent vector T ( s ) is given by
The orthic triangle is closely related to the tangential triangle, constructed as follows: let L < sub > A </ sub > be the line tangent to the circumcircle of triangle ABC at vertex A, and define L < sub > B </ sub > and L < sub > C </ sub > analogously.
* dℓ is an infinitesimal element ( a differential ) of the curve C ( i. e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C )
The aerodynamic efficiency has a maximum value, E < sub > max </ sub >, respect to C < sub > L </ sub > where the tangent line from the coordinate origin touches the drag coefficient equation plot.
Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth or non-singular, or else singular.
The deformation theory of Kunihiko Kodaira and D. C. Spencer identifies deformations via H < sup > 1 </ sup >( T ), where T is the tangent bundle sheaf K *.
If p / q is between 0 and 1, the Ford circles that are tangent to C can be described variously as

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