Posts Tagged ‘curves’
Tuesday, October 6th, 2009
How to Find Oblique Asymptotes
Description
A detailed tutorial on how to find oblique asymptotes. Step by step tutorial including several examples of how to find oblique asymptotes for reference.
Overview
There are several different types of asymptotes. In this tutorial, we will be discussing oblique asymptotes. In order to find the oblique asymptotes of a function, you must first determine if the asymptote slants. If the numerator of a rational function has exactly one degree greater than the denominator, then the function slants and therefore has an oblique asymptote. When you divide the numerator and the denominator, the term or polynomial you get is the oblique asymptote.
Tags: algebra, asymptote, asymptotes, closer, curves, degree, denominator, distance, farther, function, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, numerator, oblique, origin, polynomial, positive, slant, straight, vertical, zero
Posted in Algebra | No Comments »
Thursday, October 1st, 2009
An Overview of Bézier Curves
Description
A detailed tutorial on Bézier curves. Step by step tutorial including several examples of when and how to use Bézier curves for reference.
Overview
A Bézier curve is any parametric curve. They are extremely important in animation and computer graphic. Bézier curves can be linear, quadratic, and cubic. When Bézier curves are linear, they are expressed by the equation ![\mathbf{B}(t)=\mathbf{P}_0 + t(\mathbf{P}_1-\mathbf{P}_0)=(1-t)\mathbf{P}_0 + t\mathbf{P}_1 \mbox{ , } t \in [0,1]](http://s.wordpress.com/latex.php?latex=%5Cmathbf%7BB%7D%28t%29%3D%5Cmathbf%7BP%7D_0%20%2B%20t%28%5Cmathbf%7BP%7D_1-%5Cmathbf%7BP%7D_0%29%3D%281-t%29%5Cmathbf%7BP%7D_0%20%2B%20t%5Cmathbf%7BP%7D_1%20%5Cmbox%7B%20%2C%20%7D%20t%20%5Cin%20%5B0%2C1%5D&bg=ffffff&fg=000000&s=0 "\mathbf{B}(t)=\mathbf{P}_0 + t(\mathbf{P}_1-\mathbf{P}_0)=(1-t)\mathbf{P}_0 + t\mathbf{P}_1 \mbox{ , } t \in [0,1]")
. This is equivalent to linear interpolation.When Bézier curves are quadratic, they are expressed by the equation ![\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2 \mbox{ , } t \in [0,1].](http://s.wordpress.com/latex.php?latex=%5Cmathbf%7BB%7D%28t%29%20%3D%20%281%20-%20t%29%5E%7B2%7D%5Cmathbf%7BP%7D_0%20%2B%202%281%20-%20t%29t%5Cmathbf%7BP%7D_1%20%2B%20t%5E%7B2%7D%5Cmathbf%7BP%7D_2%20%5Cmbox%7B%20%2C%20%7D%20t%20%5Cin%20%5B0%2C1%5D.&bg=ffffff&fg=000000&s=0 "\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2 \mbox{ , } t \in [0,1].")
. They are also known as parabolic segments. When Bézier curves are cubic, they are expressed by the equation ![\mathbf{B}(t)=(1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3 \mbox{ , } t \in [0,1].](http://s.wordpress.com/latex.php?latex=%5Cmathbf%7BB%7D%28t%29%3D%281-t%29%5E3%5Cmathbf%7BP%7D_0%2B3%281-t%29%5E2t%5Cmathbf%7BP%7D_1%2B3%281-t%29t%5E2%5Cmathbf%7BP%7D_2%2Bt%5E3%5Cmathbf%7BP%7D_3%20%5Cmbox%7B%20%2C%20%7D%20t%20%5Cin%20%5B0%2C1%5D.&bg=ffffff&fg=000000&s=0 "\mathbf{B}(t)=(1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3 \mbox{ , } t \in [0,1].")
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Tags: algebra, algorithm, Bézier, curve, curves, Math, paths, Pierre Bézier
Posted in Algebra | No Comments »
Tuesday, September 29th, 2009
Definition of a Hypocycloid
Description
A detailed tutorial on the definition of a hypocycloid. Step by step tutorial including a visual example of the definition of a hypocycloid for reference.
Overview
A hypocycloid is not really an equation, or a graph, or any true function. A hypocycloid is simply a representation of the edge of a wheel or other circular item rolling on the inside of a circle to form curves. What is more noticeable than the curves it forms is the shape enclosed by the curves, which is almost like a stretched out diamond. This stretched out shape is the real hypocycloid.
Tags: brachistochrone problem, Calculus, circle, circular wheel, curves, cycloid, epicycloid, hypocycloid, Math, parameter, polar coordinates, polar graph, radians, roulette, round, tautochrone problem, The Helen of Geometers
Posted in Calculus | No Comments »
Tuesday, September 29th, 2009
Definition of an Epicycloid
Description
A detailed tutorial on the definition of an epicycloid. Step by step tutorial including a visual example of the definition of an epicycloid for reference.
Overview
An epicycloid is not really an equation, or a graph, or any true function. An epicycloid is simply a representation of the edge of a wheel or other circular item rolling along the edge of a circle to form curves. The curve it forms is really several concave down curves side by side, in a circular pattern.
Tags: brachistochrone problem, Calculus, circle, circular wheel, curves, cycloid, epicycloid, hypocycloid, Math, parameter, polar coordinates, polar graph, radians, roulette, round, tautochrone problem, The Helen of Geometers
Posted in Calculus | No Comments »
Tuesday, September 29th, 2009
How to Find Horizontal Asymptotes
Description
A detailed tutorial on how to find horizontal asymptotes. Step by step tutorial including several examples of how to find horizontal asymptotes for reference.
Overview
There are several different types of asymptotes. In this tutorial, we will be discussing horizontal asymptotes. In order to find the horizontal asymptotes of a function, take the limit of the function to infinity. Every function has a horizontal asymptote if it has a limit to infinity. The limit is your horizontal asymptote.
Tags: algebra, asymptotes, closer, curves, distance, farther, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, oblique, origin, postive, straight, vertical, zero
Posted in Algebra | No Comments »
Tuesday, September 29th, 2009
How to Find Vertical Asymptotes
Description
A detailed tutorial on how to find vertical asymptotes. Step by step tutorial including several examples of how to find vertical asymptotes for reference.
Overview
There are several different types of asymptotes. In this tutorial, we will be discussing vertical asymptotes. In order to find the vertical asymptotes of a function, we must first determine if there is a vertical asymptote. There is only a vertical asymptote if the limit of the function is equal to positive or negative infinity. If that is true, then the limit will reveal the vertical asymptote.
Tags: algebra, asymptotes, closer, curves, distance, farther, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, oblique, origin, postive, straight, vertical, zero
Posted in Algebra | No Comments »
Tuesday, September 29th, 2009
Introduction to Asymptotes
Description
A detailed tutorial on how to find asymptotes. Step by step tutorial including several examples of how to find asymptotes for reference.
Overview
An asymptote of a curve is a way of describing the behavior of the curve above the origin by comparing it to another curve. The second curve is considered an asymptote of the first if the distance between the two approaches zero as the points themselves extend to infinity. Another way of describing this is that the first curve gets closer to the second as it gets farther from the origin. If the asymptote is a straight line, it is called a linear asymptote.
Tags: algebra, asymptotes, closer, curves, distance, farther, horizontal, linear, lines, Math, nonlinear, oblique, origin, straight, vertical, zero
Posted in Algebra | No Comments »
