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Consider, for example, the one-parameter family of tangent lines to the parabola y = x 2. These are given by the generating family F(t,(x,y)) = t 2 – 2tx + y. The zero level set F(t 0,(x,y)) = 0 gives the equation of the tangent line to the parabola at the point (t 0,t 0 2).
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
The coefficient a is called the slope of the function and of the line (see below). If the slope is a = 0 {\displaystyle a=0} , this is a constant function f ( x ) = b {\displaystyle f(x)=b} defining a horizontal line, which some authors exclude from the class of linear functions. [ 3 ]
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
A non-vertical line can be defined by its slope m, and its y-intercept y 0 (the y coordinate of its intersection with the y-axis). In this case, its linear equation can be written = +. If, moreover, the line is not horizontal, it can be defined by its slope and its x-intercept x 0. In this case, its equation can be written
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables at 1+n(n+3)/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted ...
The involutes of a given curve are a set of parallel curves. For example: the involutes of a circle are parallel spirals (see diagram). And: [17] A parabola has as (two-sided) offsets rational curves of degree 6. A hyperbola or an ellipse has as (two-sided) offsets an algebraic curve of degree 8.
A curve, which intersects any curve of a given pencil of (planar) curves by a fixed angle is called isogonal trajectory. Between the slope η ′ {\displaystyle \eta '} of an isogonal trajectory and the slope y ′ {\displaystyle y'} of the curve of the pencil at a point ( x , y ) {\displaystyle (x,y)} the following relation holds: