Slope-intercept form linear equations Video transcript A line goes through the points -1, 6 and 5, 4.
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. 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.Write an equation for the line that passes through the points (1,6) and (3,-4).
We know that we are given two points. Let's use a chart to help us organize our information. You can find the straight-line equation using the point-slope form if they just give you a couple points: Find the equation of the line that passes through the points (–2, 4) and (1, 2). I've already answered this one, but let's look at the process.
In the last lesson, I showed you how to get the equation of a line given a point and a slope using the formula Anytime we need to get the equation of a line, Let's find the equation of the line that passes through the points.
This one's a two-stepper STEP 1: Find the .
Write an equation for the line that passes through the points (1,6) and (3,-4). We know that we are given two points.
Let's use a chart to help us organize our information. In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form = + + which is non-singular; that is, the curve has no cusps or self-intersections.
(When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to comprise all non-singular cubic curves; see § Elliptic curves over a general field below.).