Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given this basic definition, further criteria are needed to establish a dividing line between postlexical clitics and morphological affixes, since both are characterized by a lack of prosodic autonomy.

: Given two points, determine the azimuth and length of the line ( straight line, arc or geodesic ) that connects them.

Given a function f of a real variable x and an interval of the real line, the definite integral

# Given any two distinct points, there is exactly one line incident with both of them.

Given ailing William III's reluctance to remarry, inclusion of Sophia in the line of succession was becoming more likely.

Given India's historic position that the original intent of the line was to separate the two nations by the highest mountains in the world, in these locations India extended its forward posts northward to the ridges, regarding this move as compliant with the original border proposal, although the Simla Convention did not explicitly state this intention.

Given the consensus that " the most archaic layer of the biography, that of the stories of the kussās popular story-tellers " ( Sīra, EI² ), this may prove a fruitful line of inquiry.

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.

Given two points P and Q on C, let s ( P, Q ) be the arc length of the portion of the curve between P and Q and let d ( P, Q ) denote the length of the line segment from P to Q.

Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line.

Hypothesis: Given AD is a straight line, and AB, BC, and CD are all equal length,

* Given two points, to draw the line connecting them.

Given that the two endpoints of the line segment are specified at positions ( x1, y1 ) and ( x2, y2 ).

Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation.

Given the death of Princess Charlotte without surviving children, should the Ord or Lowe links be substantiated, the line descended through them would join a large number of claimed surviving descendants of King George IV.

Given a point in the plane, draw a straight line connecting it with the north pole on the sphere.

" Given that the parched region of the Gran Chaco ( largely uninhabited ) had been under dispute between Bolivia and Paraguay ever since the creation of both republics, each proceeded to establish a line of small garrisons ( fortines ), simply to establish a national presence and press their claims.

# Given a point p < sub > 1 </ sub > and a line l < sub > 1 </ sub >, there is a unique fold perpendicular to l < sub > 1 </ sub > that passes through point p < sub > 1 </ sub >.

# Given two points p < sub > 1 </ sub > and p < sub > 2 </ sub > and a line l < sub > 1 </ sub >, there is a fold that places p < sub > 1 </ sub > onto l < sub > 1 </ sub > and passes through p < sub > 2 </ sub >.

Given a point p < sub > 1 </ sub > and a line l < sub > 1 </ sub >, there is a unique fold perpendicular to l < sub > 1 </ sub > that passes through point p < sub > 1 </ sub >.

Given two points p < sub > 1 </ sub > and p < sub > 2 </ sub > and a line l < sub > 1 </ sub >, there is a fold that places p < sub > 1 </ sub > onto l < sub > 1 </ sub > and passes through p < sub > 2 </ sub >.

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

# Given two lines l < sub > 1 </ sub > and l < sub > 2 </ sub >, there is a fold that places l < sub > 1 </ sub > onto l < sub > 2 </ sub >.

# Given two points p < sub > 1 </ sub > and p < sub > 2 </ sub > and two lines l < sub > 1 </ sub > and l < sub > 2 </ sub >, there is a fold that places p < sub > 1 </ sub > onto l < sub > 1 </ sub > and p < sub > 2 </ sub > onto l < sub > 2 </ sub >.

# Given one point p and two lines l < sub > 1 </ sub > and l < sub > 2 </ sub >, there is a fold that places p onto l < sub > 1 </ sub > and is perpendicular to l < sub > 2 </ sub >.

Given two lines l < sub > 1 </ sub > and l < sub > 2 </ sub >, there is a fold that places l < sub > 1 </ sub > onto l < sub > 2 </ sub >.

: Given a point ( in terms of its coordinates ) and the direction ( azimuth ) and distance from that point to a second point, determine ( the coordinates of ) that second point.

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.

Given an operator on Hilbert space, consider the orbit of a point under the iterates of.

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.

: Given a determination as to the governing jurisdiction, a court is " bound " to follow a precedent of that jurisdiction only if it is directly in point.

# Given any two distinct lines, there is exactly one point incident with both of them.

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.

* Given two hypothetical point particles each of Planck mass and elementary charge, separated by any length, α is the ratio of their electrostatic repulsive force to their gravitational attractive force.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given the broad expanse of the Malvasia family, generalizations about the Malvasia wine are difficult to pin point.

At that point, two wealthy Pittsburgh, Pennsylvania businessmen, Hay Walker, Jr., and Thomas H. Given, financed the formation of National Electric Signaling Company ( NESCO ) to carry on Fessenden's research.

Given an inertial frame of reference and an arbitrary epoch ( a specified point in time ), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

* Given a point and a circle, to draw either tangent.

Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.