Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of " -sphere," with geometers referring to the number of coordinates in the underlying space and topologists referring to the dimension of the surface itself Coxeterp. As a result, geometers call the circumference of the usual circle the 2-sphere, while topologists refer to it as the 1-sphere and denote it. The circle is a conic section obtained by the intersection of a cone with a plane perpendicular to the cone 's symmetry axis.
Lines through a sphere A line can intersect a sphere at one point in which case it is called a tangent. It can not intersect the sphere at all or it can intersect the sphere at two points, the entry and exit points.
For the mathematics for the intersection point s of a line or line segment and a sphere see this. Antipodal points A line that passes through the center of a sphere has two intersection points, these are called antipodal points.
Planes through a sphere A plane can intersect a sphere at one point in which case it is called a tangent plane. Otherwise if a plane intersects a sphere the "cut" is a circle.
Lines of latitude are examples of planes that intersect the Earth sphere. Lines of latitude Lines of longitude Meridians Great Circles A great circle is the intersection a plane and a sphere where the plane also passes through the center of the sphere.
Lines of longitude and the equator of the Earth are examples of great circles. Two points on a sphere that are not antipodal define a unique great circle, it traces the shortest path between the two points.
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If the points are antipodal there are an infinite number of great circles that pass through them, for example, the antipodal points of the north and south pole of Earth there are of course infinitely many others.
Great circles define geodesics for a sphere. A geodesic is the closest path between two points on any surface. Lune A lune is the area between two great circles who share antipodal points. Unlike a plane where the interior angles of a triangle sum to pi radians degreeson a sphere the interior angles sum to more than pi.
As the sphere becomes large compared to the triangle then the the sum of the internal angles approach pi. Calculating Centre Two lines can be formed through 2 pairs of the three points, the first passes through the first two points P1 and P2.
Line b passes through the next two points P2 and P3. The equation of these two lines is where m is the slope of the line given by The centre of the circle is the intersection of the two lines perpendicular to and passing through the midpoints of the lines P1P2 and P2 P3.
Alternatively one can also rearrange the equations of the perpendiculars and solve for y. Radius The radius is easy, for example the point P1 lies on the circle and we know the centre The denominator mb - ma is only zero when the lines are parallel in which case they must be coincident and thus no circle results.
If either line is vertical then the corresponding slope is infinite. This can be solved by simply rearranging the order of the points so that vertical lines do not occur. Equation of a Sphere from 4 Points on the Surface Written by Paul Bourke June Given 4 points in 3 dimensional space [ x1,y1,z1 x2,y2,z2 x3,y3,z3 x4,y4,z4 ] the equation of the sphere with those points on the surface is found by solving the following determinant.Great Circles.
A great circle is the intersection a plane and a sphere where the plane also passes through the center of the sphere. Lines of longitude and the . Circle equation calculator This calculator can find the center and radius of a circle given its equation in standard or general form.
Also, it can find equation of a circle given its center and radius. Find the equation of a circle whose center is at the point (-2, 3) and its diameter has a length of Find the center and the radius of the circle whose equation is given by x 2 + 4x + y 2 - 8y = 5.; Find an equation of the circle that is is tangent to both the x and y axes, with a radius of 4 and whose center is located in the second quadrant.
Improve your math knowledge with free questions in "Write equations of circles in standard form using properties" and thousands of other math skills.
In thermodynamics, the Gibbs free energy (IUPAC recommended name: Gibbs energy or Gibbs function; also known as free enthalpy to distinguish it from Helmholtz free energy) is a thermodynamic potential that can be used to calculate the maximum of reversible work that may be performed by a thermodynamic system at a constant temperature and pressure (isothermal, isobaric).
A Time-line for the History of Mathematics (Many of the early dates are approximates) This work is under constant revision, so come back later. Please report any errors to me at [email protected]