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Parametric Equation of a Circle
I don't get the " the curved 2-dimensional surface only—not the top, bottom or solid inside " part of the question. Is z just equal to z, since the cylinder's height depends on z? Is my parametrization correct? Sign up to join this community.
The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Parametrizing a Cylinder using Cylindrical Coordinates? Ask Question. Asked 3 years, 3 months ago. Active 1 year, 10 months ago. Viewed 7k times. Otherwise, you're good. Active Oldest Votes. Emilio Novati Emilio Novati Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.
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Thread starter dagar Start date Jan 14, Any guidance on how to find this intersection in a parameterized form would be most appreciated. Googling hasn't turned up anything particularly usefull, but a few scattered examples. I was also wondering if anyone knew of useful sites with this information. Insights Author. Gold Member. Any guidance on how to find this intersection in a parameterized form would be most appreciated You must log in or register to reply here. Related Threads on Intersection of cylinder and plane Intersection of a plane and cylinder.Surface And Flux Integrals, Parametric Surf., Divergence/Stoke's Theorem: Calculus 3 Lecture 15.6_9
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Using trigonometry, we can find the coordinates of P from the right triangle shown. In this triangle the radius r is the hypotenuse. To see why this is, recall that in a right triangle, the sine of an angle is the opposite side divided by the hypotenuse. In the figure on the right. In the applet above, the side opposite t has a length of ythe y coordinate of P. The hypotenuse is the radius r. Therefore Multiply both sides by r. By similar means we find that The parametric equation of a circle From the above we can find the coordinates of any point on the circle if we know the radius and the subtended angle.
Then we just add or subtract fixed amounts to the x and y coordinates. In the figure above, drag the center point C to see this. In the above equations, the angle t theta is called a 'parameter'.
This is a variable that appears in a system of equations that can take on any value unless limited explicitly but has the same value everywhere it appears. A parameter values are not plotted on an axis. This form of defining a circle is very useful in computer algorithms that draw circles and ellipses.
In fact, all the circles and ellipses in the applets on this site are drawn using this equation form. For more on this see An Algorithm for Drawing Circles. Using the Pythagorean Theorem to solve the triangle in the figure above we get the more common form of the equation of a circle. For more see Basic equation of a circle and General equation of a circle.
To demonstrate that these forms are equivalent, consider the figure below. In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off. Home Contact About Subject Index.Parametric surfaces render very slowly in POV 3. Sometimes it is difficult, or even impossible, to specify a single equation in x, y and z that specifies the surface but a set of parametric equations might be available.
Each value of theta gives a value for x and y, i. As theta varies from 0 to 2pi the point goes round the unit circle. In the 3d case we need three parametric equations, one each for x, y and z; and we need two parameters which we will call u and v.
As u varies from 0 to 2pi, the point goes round a circle. As v varies from -2 to 2 the point moves parallel to the z axis. It turns out that these are the parametric equations for a cylinder.
This is the cylinder generated by these parametric equations. Here's a sort of conical spiral that would be very difficult to specify without using parametric equations. This is the same surface as last time, but in this case I've declared two of the equations beforehand. It is possible to perform variable substitution here.
The "precompute" keyword can speed up the rendering by telling the renderer to store some calculations in an array, thus trading memory and parsing time against rendering time. I find that "precompute 18, x,y,z" tends to give a reasonable speed improvement on my machine.
Higher values cause the parse time to become rather long. In this case, there's no speed gain from precomputing y, but there's not much of a penalty either.
Parametric equations for cylinder/plane intersection
I get the distinct impression that things like pigment functions aren't really supported with parametric surfaces. The parser seems to be trying to prevent me from using "pigment" in the same function as "u" or "v", but the following syntax is accepted. This is the Astroidal Ellipse. The surface goes in where an ordinary ellipse goes out. This surface is called the Bohemian Dome.
I want a parametric equation of a sine wave at a small ramp angle wrapped around a cylindrical body 3D. The parametric equation below gets me close to what I'm looking for, but not quite since the sine wave itself is not rotated peaks still in line with the longitudinal direction of the cylindrical body. In 2D, the following parametric equation will give me a rotated sine wave.
How can I convert this to 3D around the cylinder following a helical pattern? The theta variable is the angle from the x-axis that the sine wave is rotated.
Fancy animated picture! Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Parametric Equation of sine wave helically wrapped around a cylinder Ask Question. Asked 5 years, 10 months ago. Active 5 years, 10 months ago. Viewed 2k times. I'm not looking for a toroid in a helical pattern, if that is what you are asking. The first parametric equation gives me a sine wave in which all the peaks are aligned in the longitudinal direction.
Active Oldest Votes. Thank you. Can you explain how you got this?This equation is very similar to the one used to define a circle, and much of the discussion is omitted here to avoid duplication. See Parametric equation of a circle as an introduction to this topic. The only difference between the circle and the ellipse is that in a circle there is one radius, but an ellipse has two:.
Note that the equations on this page are true only for ellipses that are aligned with the coordinate plane, that is, where the major and minor axes are parallel to the coordinate system. In the applet above, drag one of the four orange dots around the ellipse to resize it, and note how the equations change to match.
Just as with the circle equationswe add offsets to the x and y terms to translate or "move" the ellipse to the correct location. In the applet above, drag the orange dot at the center to move the ellipse, and note how the equations change to match.
Also, adjust the ellipse so that a and b are the same length, and convince yourself that in this case, these are the same equations as for a circle.
In the applet above, drag the right orange dot left until the two radii are the same. This is a circle, and the equations for it look just like the parametric equations for a circle. This demonstrates that a circle is just a special case of an ellipse. The parameter t can be a little confusing with ellipses.
For any value of tthere will be a corresponding point on the ellipse. But t is not the angle subtended by that point at the center. To see why this is so, consider an ellipse as a circle that has been stretched or squashed along each axis.
In the figure below we start with a circle, and for simplicity give it a radius of one a " unit circle ". The angle t defines a point on the circle which has the coordinates The radius is one, so it is omitted.
The blue ellipse is defined by the equations So to get the corresponding point on the ellipse, the x coordinate is multiplied by two, thus moving it to the right. This causes the ellipse to be wider than the circle by a factor of two, whereas the height remains the same, as directed by the values 2 and 1 in the ellipse's equations. So as you can see, the angle t is not the same as the angle that the point on the ellipse subtends at the center.
Using the Pythagorean Theorem to find the points on the ellipse, we get the more common form of the equation. For more see General equation of an ellipse. This form of defining an ellipse is very useful in computer algorithms that draw circles and ellipses.
In fact, all the circles and ellipses in the applets on this site are drawn using this equation form. For more on this see An Algorithm for Drawing Circles.The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder a. A cylinder of this sort having a polygonal base is therefore a prism Zwillingerp. Harris and Stockerp. Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself Zwillingerp.
To make matters worse, according to topologists, a cylindrical surface is not even a true surfacebut rather a so-called surface with boundary Henlepp. As if this were not confusing enough, the term "cylinder" when used without qualification commonly refers to the particular case of a solid of circular cross section in which the centers of the circles all lie on a single line i. A cylinder is called a right cylinder if it is "straight" in the sense that its cross sections lie directly on top of each other; otherwise, the cylinder is said to be oblique.
The unqualified term "cylinder" is also commonly used to refer to a right circular cylinder Zwillingerp. The right cylinder of radius with axis given by the line segment with endpoints and is implemented in the Wolfram Language as Cylinder [ x1y1z1x2y2z2r ].
The illustrations above show a circular right cylinder of height and radius. If a plane inclined with respect to the caps of a right circular cylinder intersects a cylinder, it does so in an ellipse. The cylinder was extensively studied by Archimedes in his two-volume work On the Sphere and Cylinder in ca. As illustrated above, a cylinder can be described topologically as a square in which top and bottom edges are given parallel orientations and the left and right edges are joined to place the arrow heads and tails into coincidence Graypp.
The cylindrical surface of a circular cylinder has Euler characteristic 0 Alexandroffp. The lateral surface of a cylinder of height and radius can be described parametrically by.
These are the basis for cylindrical coordinates. The lateral surface area and volume of the cylinder of height and radius are. The formula for the volume of a cylinder leads to the mathematical joke: "What is the volume of a pizza of thickness and radius?
This result is sometimes known as the second pizza theorem. The interior of the cylinder of radiusheightand mass has moment of inertia tensor about its centroid is. The fact that. It is possible to arrange seven finite cylinders so that each is tangent to the other six, as illustrated above. Alexandroff, P. Combinatorial Topology. New York: Dover, Beyer, W.
Harris, J. New York: Springer-Verlag, pp. Henle, M. A Combinatorial Introduction to Topology. Hilbert, D. New York: Chelsea, pp. Kern, W.