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Curved line segment is drawn
- When two points on a plane surface are joined, a straight line segment is obtained. But if two points in a curved surface is joined, a curved line segment is drawn. When a line segment is extended towards both sides unlimitedly it becomes a line.
www.math-only-math.com/points-and-line-segment.htmlPoints and Line Segment | Two Points in a Curved Surface ...
One involves the effects of gravitation. Another involves the ideas of space-time which we already studied. The third involves the idea of curved space-time. We will simplify our subject in the beginning by not worrying about gravity and by leaving out the time—discussing just curved space.
When two points on a plane surface are joined, a straight line segment is obtained. But if two points in a curved surface is joined, a curved line segment is drawn. When a line segment is extended towards both sides unlimitedly it becomes a line.
On a curved surface, the shortest distance between two points is actually a curve, technically known as a geodesic, which we can perhaps visualize when we think, for example, of a plane flying the shortest route between London and New York which, as travelers will know, follows a "great circle" path over Newfoundland rather than what appears to ...
Some solids like cylinder have both plane and curved surfaces. 3-D shaped solids are covered by 2-D surfaces. If any two points are joined, a line segment is formed. In this chapter planes and lines will be discussed.
- Unfamiliar Geometries Become Familiar
- The New Geometry of 5None Is Spherical Geometry
- Is The Geometry of 5None Consistent?
- A Triangle in The Geometry of 5more
- Corrections to The Other Postulates
In the last chapter, we explored the geometry induced by the postulate 5NONEby means of the traditional construction techniques of geometry familiar to Euclid. We drew lines and found points only as allowed by the various postulates. The outcome was a laborious construction of circles and triangles with some quite peculiar properties. We constructe...
The geometry of 5NONE proves to be very familiar; it is just the geometry that is natural to the surface of a sphere, such as is our own earth, to very good approximation. The surface of a sphere has constant curvature. That just means that the curvature is everywhere the same. To see how the connection to the geometry of 5NONEworks, we need only i...
How do we know that a more imaginative, more thorough analysis might not eventually produce a contradiction? That is, how do we knowthat the new geometry is consistent? The question could be answered by a proof of the consistency of the geometry. Alas, advances in twentieth century mathematics have shown that proving the consistency of a rich syste...
It is worth taking a moment to see how at least one of these results follows from the replacing of the postulate of 5ONE with 5MORE. We shall see how it follows that the sum of the angles of a triangleturns out to be less that 180 degrees. To begin, we consider a straight line AB. At A we erect a perpendicular and mark any point O on the perpendicu...
The general conception is that we can keep the first four postulates and generate one of the three geometries of 5NONE, 5ONE or 5MORE merely by adopting the appropriate form of the fifth postulate. This general conception is correct but it needs a small technical correction. It arises through the identification of the geometry of 5NONE as the geome...
Those aren't straight lines though, they're geodesics. A straight line is the shortest path between two points, a geodesic is the shortest path between two points along the surface of a sphere.
People also ask
What happens if two points are joined in a curved surface?
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In the first case, when the vessel is empty, light from the arrow refracts at the curved interface, moves through the glass and enters into air then it again undergoes refraction on the opposite curved surface of the vessel (at the other end from where we are looking) and comes out into the air.
