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.

Then rotate the tangent screw back and forth so that the reflected image passes alternately above and below the direct view.

Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.

Then consider a vector tangent to:

Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )

Then each tangent space is just R. On each copy of R at the point y, we introduce the modified inner product

Then a connection on the tangent bundle of M, called an affine connection, distinguishes a class of curves called ( affine ) geodesics.

Then, multiply the tangent by the front wheel braking effort percentage and divide by the ratio of the center of gravity height to the wheelbase.

Then, if is finite-dimensional, then so is the submanifold ; likewise, the tangent space has the same dimension as.

Then the differential of φ, φ < sub >*</ sub > = dφ ( or Dφ ), is a vector bundle morphism ( over M ) from the tangent bundle TM of M to the pullback bundle φ < sup >*</ sup > TN.

Then the inner product of two tangent vectors is

Then, hit the inverse tangent key ( or tan < sup >- 1 </ sup >) and the answer just calculated.

Then the unit tangent vector T may be written as

Let C ′ be the curve obtained by shrinking C by a factor of 2 toward P. Then the point R ′ corresponding to R is the center of the rectangle PXRY, and the tangent to C ′ at R ′ bisects this rectangle parallel to PY and XR.

Then all the lines which are common tangents to both P = 0 and Q = 0 are tangent to C. So, by the AF + BG theorem, the tangential equation of C has the form HP + KQ = 0.

Then, integration of the tangent field ( done numerically, if not analytically ) yields the curve.

Then a vector field H on TM ( that is, a section of the double tangent bundle TTM ) is a semispray on M, if any of the three following equivalent conditions holds:

Fix a point p ∈ M, and an orthonormal basis X < sub > 1 </ sub >, X < sub > 2 </ sub > of tangent vectors at p. Then the principal curvatures are the eigenvalues of the symmetric matrix

Then at time the particle acquires the velocity, as if it underwent an elastic push from the infinitely-heavy plane, which is tangent to at the point, and at time moves along the normal to at with the velocity.

The key tool is the curve X given by the set of pairs ( p, ℓ ) where p is on the conic C and ℓ is tangent to the conic D. Then X is smooth ; more specifically X is an elliptic curve.

Then construct two additional circles, each tangent to AM, BC, and to the circumcircle.

Then a Taylor cone is formed under the application of a strong electric field.

Then about 1, 100 years ago several dacitic domes, the Chaos Crags, protruded through these cones and obliterated all but half of the southernmost cone.

Then the “ jet, filled with smoke and burning shreds becomes a whirling inverted cone flashing with thousands of yellow sparks in a brilliant pyrotechnic display ”.

Then K ( A ) is a triangulated category ; the distinguished triangles consist of triangles isomorphic to a morphism with its mapping cone ( in the sense of chain complexes ).

Then the mapping cone C < sub > f </ sub > is homeomorphic to two disks joined on their boundary, which is topologically the sphere S < sup > 2 </ sup >.

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Then X is reflexive if and only if each X < sub > j </ sub > is reflexive.

Then X is separable if and only if X ′ is separable.

Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).

Then all elements of X equivalent to each other are also elements of the same equivalence class.

Then the quotient space X /~ can be naturally identified with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.

Then the expectation of this random variable X is defined as

Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.

Then the joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

Then for a specific value x of X, the function L ( θ | x )

Then the observation that X

Then the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.

Then X has cardinality at most and cardinality at most if it is first countable.

Then A is dense in C ( X, R ) if and only if it separates points.

Then ρ will be the finest completely regular topology on X which is coarser than τ.