Posts Tagged ‘curve’
Thursday, October 8th, 2009
Introduction to Inflection Points
Description
A detailed tutorial on inflection points. Step by step tutorial including several examples of inflection points and how to locate inflection points for reference.
Overview
An inflection point, sometimes also known as a point of inflection, is a point on the graph of a function at which the function changes sign. This means that a concave up curve will become a concave down curve, or a concave down curve will become a concave up curve. Inflection points are also points of local maxima and local minima of a function. There are two ways to categorize inflection points. There are stationary points of inflection, and non-stationary points of inflection. Stationary points are formed when the function is zero, and non-stationary points are when the function is not zero.
Tags: Calculus, concave, curve, down, function, inflection, inflexion, local, Math, maxima, minima, non-stationary, point, saddle-point, sign, stationary, up
Posted in Calculus | No Comments »
Thursday, October 8th, 2009
How to Use the Second Derivative Test
Description
A detailed tutorial on how to use the second derivative test. Step by step tutorial including several examples of how to use the second derivative test for reference.
Overview
The second derivative test is more well-known than the first derivative test, and is often thought to be more accurate. The second derivative test states that if the second derivative of a function is less than zero, then there is a local maximum at x. If the second derivative of a function is greater than zero, then there is a local minimum at x. However, if the second derivative of a function is equal to zero, then the local maximum or minimum cannot be determined. Then you must use the first derivative test to figure it out. The second derivative test can also be used to figure out the concavity of a function – that is, if a curve is pointing up or down. This is normally used to help create the image of the function on a graph.
Tags: Calculus, chart, concavity, critical points, curve, derivative, equals, extrema, extremum, first derivative test, function, graph, Math, maxima, maximum, minima, minimum, negative, positive, second derivative test
Posted in Calculus | No Comments »
Tuesday, October 6th, 2009
Definition of a Cornu Spiral
Description
A detailed tutorial on Cornu spirals. Step by step tutorial including a visual example of a Cornu spiral for reference.
Overview
The Cornu spiral is also known as a Euler spiral and clothoid. It is generated as a straight line that branches out, and then turns up on one end and down on the other, both spiraling into tight curls. It is formed by a parametric plot of S(t) against C(t). They are very closely linking to Fresnal integrals and have been sometimes thought of as a solution.
Tags: Calculus, clothoid, Cornu spiral, curl, curve, differential equations, Euler spiral, Fresnal integral, Geometry, line, Math, parametric, plot, solution, spiral
Posted in Differential Equations | No Comments »
Thursday, October 1st, 2009
Definition of a Cissoid
Description
A detailed tutorial of the definition of a cissoid. Step by step tutorial including a visual example of the definition of a cissoid for reference.
Overview
A cissoid is a curve that is found from two curves and a point. The point is known as “the pole”. Each definition of a cissoid and instructions on how to find the cissoid are a little different, depending on your source. However, the basic definition of a cissoid is that it is an average point that is relative to the pole and found between the two given curves. Keep in mind that the cissoid is a curve, not a point, but the curve is to be drawn from the point that is found.
Tags: average, cissoid, curve, Geometry, line, locus, Math, point, polar coordinates, pole, relative
Posted in Geometry | 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 Hyperbola
Description
A detailed tutorial of the definition of a hyperbola. Step by steo tutorial including several examples of the definition of a hyperbola for reference.
Overview
A hyperbola is similar to a parabola, but there is one difference – the hyperbola has two branches. You can think of it in the 2D form as a concave up parabola on top of a concave down parabola. Many people refer to the hyperbola as the “bow” because that is what it resembles. Like the parabola, a hyperbola is caused by the intersection of a conical surface and a plane.
Tags: concave, conic, conical surface, curve, focus, Geometry, graph, hyperbola, intersect, Math, parabola, plane
Posted in Geometry | No Comments »
Tuesday, September 29th, 2009
Definition of a Parabola
Description
A detailed tutorial of the definition of a parabola. Step by step tutorial including a visual example of the definition of a parabola for reference.
Overview
A parabola is an elongated curve that is used often in graphing. A parabola is formed by the graph of y = x^2, and its traditional form is concave up. Technically, the parabola is actually a conic section, which is the intersection of a conical surface and a plane parallel to the generated straight line of that surface.
Tags: concave, conic, conical surface, curve, focus, Geometry, graph, intersect, Math, parabola, plane, y=x^2
Posted in Geometry | No Comments »
Tuesday, September 29th, 2009
Definition of a Cardioid
Description
A detailed tutorial on the definition of a cardioid. Step by step tutorial including a visual example of a cardioid for reference. The regular cardioid shape is when there is no loop within the circle.
Overview
A limaçon is a special kind of curve that is formed when a circle rolls around the outside of a circle of equal radius. The cardioid is a unique type of limaçon, where the point generating the curve lies on the rolling circle, resulting in a cusp. The cardioid is named for its resemblance to a heart.
Tags: Calculus, cardioid, circle, curve, cusp, limaçon, Math, polar coordinates, polar graph, rolling circle
Posted in Calculus | No Comments »
Thursday, September 17th, 2009
How to Solve the Equation of a Tangent Line
Description
A detailed tutorial on the solving of the equation of a tangent line. Step by step tutorial including several examples of how to solve the equation of a tangent line for reference.
Overview
A tangent line is the straight line to a curve at any given point that just touches the curve at that point. In a mathematical sense, at that point the tangent line is going in the same direction as the curve. To solve the equation of a tangent line, say that the curve is the graph of the function y = f(x). The point at which the tangent line intersects the curve is p = (a, f(a)). Now, take another point on the curve that is close to the line, which can be expressed as q = (a + h, f(a + h)). The secant line passes through both of these points, and the slope of the secant line is equal to the difference quotient. The difference quotient is expressed as:
Those who have already studied limits will recognize the difference quotient to be the definition of a limit function.
Tags: Calculus, curve, equation, equation of a tangent line, function, graph, limit, line, Math, secant, slope, slope of secant line, tangent
Posted in Calculus | No Comments »
