Its parameter representation is, \( (x,y,z) = \left(r\cos\phi,r\sin\phi,R\cos\frac\phi2\right).\), Retrieved from "http://en.wikipedia.org/" All text is available under the terms of the GNU Free Documentation License. If R < r + a, the intersection of sphere and cylinder consists of a single closed curve. The intersection is the single point (r, 0, 0). The intersection is the single point (r, 0, 0). Sphere centered on cylinder axis, If the center of the sphere lies on the axis of the cylinder, a = 0. Sphere touches cylinder in one point If the sphere is smaller than the cylinder (R < r) and a+R = r, the sphere lies in the interior of the cylinder except for one point. Calculations at an intersection of sphere and cylinder. In this case, the intersection of sphere and cylinder consists of two closed curves, which are mirror images of each other. We take the radius of the sphere equal to unity and use for the two remaining variables the following symbols: (!, radius of the cylinder; The intersection is the collection of points satisfying both equations. (function() { var po = document.createElement('script'); po.type = 'text/javascript'; po.async = true; po.src = 'https://apis.google.com/js/plusone.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(po, s); })(); In the theory of analytic geometry for real three-dimensional space, the intersection between a sphere and a cylinder can be a circle, a point, the empty set, or a special type of curve. Non-trivial cases, Subtracting the two equations given above gives, Since x is a quadratic function of z, the projection of the intersection onto the xz-plane is the section of an orthogonal parabola; it is only a section due to the fact that -r < x < r. The vertex of the parabola lies at point (-b, 0, 0), where, Intersection consists of two closed curves. The above parametrization becomes, \( (x,y,z) = \left(r\cos\phi,r\sin\phi,2\sqrt{ar}\cos\frac{\phi}{2}\right),\). If the points are antipodal there are an infinite number of great circles that pass through them, … Their projection in the xy-plane are circles of radius r. Each part of the intersection can be parametrized by an angle \( \phi \): \( (x,y,z) = \left(r\cos\phi,r\sin\phi,\pm\sqrt{2a(b + r\cos\phi)}\right).\). For the analysis of this situation, assume (without loss of generality) that the axis of the cylinder coincides with the z-axis; points on the cylinder (with radius r) satisfy, We also assume that the sphere, with radius R is centered at a point on the positive x-axis, at point (a, 0, 0). The intersection of a sphere and a cylinder produces two regions in space of interest in Calculus. The first region is the set of all points inside the sphere but outside of the cylinder. Two points on a sphere that are not antipodal define a unique great circle, it traces the shortest path between the two points. In the special case a = r, R = 2r, the intersection is known as Viviani's curve. The intersection resembles a figure eight: it is a closed curve which intersects itself. Intersection of a sphere and a cylinder. Calottes are the curved ends of the sphere cylinder intersection, which before were parts of the surface of the sphere. Sphere centered on cylinder axis where \( \phi \) now goes through two full revolutions. It can be described by the same parameter equation as in the previous section, but the angle \( \phi\) must be restricted to -\( \phi_0 < \phi < +\phi_0 \), where \( \cos\phi_0 = -b/r\). A great circle is the intersection a plane and a sphere where the plane also passes through the center of the sphere. In that case, the intersection consists of two circles of radius r. These circles lie in the planes, If r = R, the intersection is a single circle in the plane z = 0. The volume of the intersection depends on three variables, viz. Its points satisfy. Lines of longitude and the equator of the Earth are examples of great circles. The curves contain the following extreme points: \( \left(-r, 0, \pm\sqrt{R^2 - (r+a)^2}\right);\quad \left(0, \pm r, \pm\sqrt{R^2 - (r-a)(r+a)}\right);\quad \left(+r, 0, \pm\sqrt{R^2 - (r-a)^2}\right).\). Free ebook http://tinyurl.com/EngMathYT How to determine where two surfaces intersect (sphere and cone). The curve contains the following extreme points: \( \left(-b, \pm\sqrt{r^2-b^2}, 0\right);\quad \left(0, \pm r, \pm\sqrt{R^2 - (r-a)(r+a)}\right);\quad \left(+r, 0, \pm\sqrt{R^2 - (r-a)^2}\right).\), Limiting case Viviani's curve as intersection of a sphere and a cylinder, In the case R = r + a, the cylinder and sphere are tangential to each other at point (r, 0, 0). . Sphere touches cylinder in one point, If the sphere is smaller than the cylinder (R < r) and a+R = r, the sphere lies in the interior of the cylinder except for one point. The cylinder goes straight through the center of the sphere and the cylinder radius must be smaller than the sphere radius. Trivial cases Sphere lies entirely within cylinder, If a+R < r, the sphere lies entirely in the interior of the cylinder. If R > r + a, the condition x < r cuts the parabola into two segments. The intersection is the empty set. This region can be generated by revolving a part of a circle about an axis. the radii of the sphere and the cylinder and the distance of the centre of the sphere to the axis of the cylinder.
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