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A plane curve is the image of any continuous function from an interval to the Euclidean plane.Intuitively, it is a set of points that could be traced out by a moving point. More specifically, smooth curves generally at least require that the function from the interval to the plane be continuously differentiable, and in some contexts are defined to require higher derivative
Excel's storage of numbers in binary format also affects its accuracy. [3] To illustrate, the lower figure tabulates the simple addition 1 + x − 1 for several values of x. All the values of x begin at the 15 th decimal, so Excel must take them into account. Before calculating the sum 1 + x, Excel first approximates x as a binary number
The point (,) is the vertex of the parabola. Pencil of confocal parabolas From the definition of a parabola , for any point P {\displaystyle P} not on the x -axis, there is a unique parabola with focus at the origin opening to the right and a unique parabola with focus at the origin opening to the left, intersecting orthogonally at the point P ...
For a point within the hyperbola the polar has no point with the hyperbola in common. (see diagram: , ). Remarks: The intersection point of two polars (for example: ,) is the pole of the line through their poles (here: ,).
Application: The 2-points–2-tangents property can be used for the construction of the tangent of a parabola at point , if , and the tangent at are given. Remark 1: The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem.
A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results: = + +. This will exactly fit a simple curve to three points.
The dilation is commutative, also given by = =. If B has a center on the origin, as before, then the dilation of A by B can be understood as the locus of the points covered by B when the center of B moves inside A. In the above example, the dilation of the square of side 10 by the disk of radius 2 is a square of side 14, with rounded corners ...
A superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points (x, y) with = [()], where p, a, and b are positive integers. This equation defines an open curve within the rectangle − a ≤ x ≤ a {\displaystyle -a\leq x\leq a} , 0 ≤ y ≤ b {\displaystyle 0\leq y\leq b} .