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The curve begins at and ends at ; this is the so-called endpoint interpolation property. The curve is a line if and only if all the control points are collinear. The start and end of the curve is tangent to the first and last section of the Bézier polygon, respectively.
Specific choices of give different types of Riemann sums: . If = for all i, the method is the left rule [2] [3] and gives a left Riemann sum.; If = for all i, the method is the right rule [2] [3] and gives a right Riemann sum.
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.. In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist.
An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing a.. b − 1 , a + 1 .. b , or a + 1 .. b − 1. Alternate-bracket notations like [a.. b) or [a.. b[are rarely used for integer intervals. [citation needed]
Interpolation with cubic splines between eight points. Hand-drawn technical drawings for shipbuilding are a historical example of spline interpolation; drawings were constructed using flexible rulers that were bent to follow pre-defined points.
With the n-th polynomial normalized to give P n (1) = 1, the i-th Gauss node, x i, is the i-th root of P n and the weights are given by the formula [3] = [′ ()]. Some low-order quadrature rules are tabulated below (over interval [−1, 1] , see the section below for other intervals).
For a strong acid-strong base titration monitored by pH, we have at any i'th point in the titration = [+] [] where K w is the water autoprotolysis constant.. If titrating an acid of initial volume and concentration [+] with base of concentration [], then at any i'th point in the titration with titrant volume ,