SUMMARY / RELATED TOPICS

In differential geometry, a geodesic is a curve representing in some sense the shortest path between two points in a surface, or more in a Riemannian manifold. It is a generalization of the notion of a "straight line" to a more general setting; the term "geodesic" comes from the science of measuring the size and shape of Earth. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle; the term has been generalized to include measurements in much more general mathematical spaces. In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature. More in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion. Geodesics are of particular importance in general relativity.

Timelike geodesics in general relativity describe the motion of free falling test particles. The shortest path between two given points in a curved space, assumed to be a differential manifold, can be defined by using the equation for the length of a curve, minimizing this length between the points using the calculus of variations; this has some minor technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from f to f along the curve equals |s−t|. Equivalently, a different quantity may be used, termed the energy of the curve. Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, in so doing will minimize its energy; the resulting shape of the band is a geodesic. It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere.

In such a case, any of these curves is a geodesic. A contiguous segment of a geodesic is again a geodesic. In general, geodesics are not the same as "shortest curves" between two points, though the two concepts are related; the difference is that geodesics are only locally the shortest distance between points, are parameterized with "constant speed". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points; the map t → t2 from the unit interval on the real number line to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant. Geodesics are seen in the study of Riemannian geometry and more metric geometry. In general relativity, geodesics in spacetime describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved spacetime.

More the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, their movement is constrained in various ways. This article presents the mathematical formalism involved in defining and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds; the article geodesic discusses the special case of general relativity in greater detail. The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles; the shortest path from point A to point B on a sphere is given by the shorter arc of the great circle passing through A and B. If A and B are antipodal points there are infinitely many shortest paths between them. Geodesics on an ellipsoid behave in a more complicated way than on a sphere. In metric geometry, a geodesic is a curve, everywhere locally a distance minimizer. More a curve γ: I → M from an interval I of the reals to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t1, t2 ∈ J we have d = v | t 1 − t 2 |.

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is equipped with natural parameterization, i.e. in the above identity v = 1 and d = | t 1 − t 2 |. If the last equality is satisfied for all t1, t2 ∈ I, the geodesic is called a minimizing geodesic or shortest path. In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in a length metric spac

Common Ground is a 2000 Showtime television film directed by Donna Deitch and written by Paula Vogel, Terrence McNally and Harvey Fierstein. In the 1950s, Dorothy Nelson joins the United States Navy where she meets the Friends of Dorothy, a code name for a group of gay and lesbian sailors. Nelson meets Billy. However, the NIS raids the nightclub, Nelson is among those servicemembers who receive a Blue discharge for "sexual perversion." Returning to Homer, she tries to restart her life as a public school teacher, but her Section 8 discharge prevents her from getting a job. When her homosexuality becomes public knowledge, her mother expels her from the house, forcing her to seek shelter at a family friend's grocery store. However, the townspeople disapprove of this arrangement, Nelson becomes homeless. An independent-minded woman named Janet at the local diner defends her against the verbal harassment and advises Nelson to go to the bohemian Greenwich Village, the only place where she might be free to be herself.

The second story flashes forward to the town of Homer in the 1970s, towards the end of the Vietnam War. There a closeted gay high school French language teacher, Mr. Roberts, has a student named Tobias Anderson, nicknamed Toby, on the verge of coming out of the closet, who he suspects wishes to confide in him. Roberts must keep his homosexuality a secret for the fear of losing his job, but his live-in boyfriend Gus pressures him to set a good example for the students by illustrating the importance of tolerance and justice. Toby visits a prostitute on the advice of his swimming coach, with the idea that she can help him "become a man", but rather instead gives him some good advice about being himself. After Toby is sexually assaulted by bullies and is discovered by Roberts, Roberts himself comes out to his students and lectures them on the evils of bias-motivated hatred. Toby leaves Homer to attend college in the big city; the final short story takes place in the present day, when a father and the townspeople have to come to terms with the fact that two men will be getting married during a commitment ceremony to be held in the town.

Ira, the father, is planning to lead a protest march against the wedding, while his son, Amos, is nervous about getting married and going against the cultural stereotype of gay men. The film ends on a positive note, with father and son reconciling and the wedding taking place as scheduled. Edward Asner as Ira Beau Bridges as Father Leon Harvey Fierstein as Don Erik Knudsen as Young Johnny Burroughs James LeGros as Amos Brittany Murphy as Dorothy Nelson Jason Priestley as Billy Helen Shaver as Janet Eric Stoltz as Johnny Burroughs Jonathan Taylor Thomas as Tobias "Toby" Anderson Steven Weber as Gil Roberts Scott McCord as Gus Caterina Scorsone as Peggy A Friend of Dorothy's was written by Paula Vogel; the plays star Brittany Murphy, Jason Priestley, Steven Weber, Jonathan Taylor Thomas, Edward Asner and James LeGros. The film contains three short stories about gay Americans during different time periods in the fictional town of Homer and their efforts to find "common ground" or respect from the heterosexual majority.

Common Ground on IMDb

The rules of the game are the same as those in the original Adventure Island, with the main new feature being the addition of an inventory system. Before the player begins a stage, he can choose which of Higgins' animal friends to bring, as well as whether or not he should bring one of the stone hammers he has accumulated; because of this, the player can no longer upgrade to shooting fireballs when he picks a second hammer. Instead, it gets added as a reserved hammer to the player's inventory; the checkpoint system has been eliminated and if the player dies in the middle of a stage, he must restart from the beginning. However, the stages are shorter than in the previous game; when the player strikes a place where a hidden egg is located, it will play a different noise that will serve as an indicator of its location. The bonus zones are now accessed by picking up a key located inside these hidden eggs; some of these keys will transport the player to an item room or a shortcut to the next island.

There are now underwater stages, as well as vertical-scrolling stages. When the player completes a stage riding a skateboard, he does not get to take it to the next stage like he could in the previous game; when the player completes a stage, he will be asked to choose one of ten spinning eggs that will give him a certain number of points. The player can now backtrack during a stage only up to a certain point; the boss of each island awaits the player on a specific stage at the beginning. However, if the player is defeated while fighting a boss, the boss will move to another area, forcing the player not only to clear his current stage, but another one in order to fight the boss. There are four types of dinosaur friends; these animal friends are summoned. The blue camptosaurus can walk on ice without sliding. Allgame editor Skyler Miller praised the game over the original its "improved graphics, a map screen, most four dinosaur buddies for Master Higgins to ride, each with their own special ability".

Adventure Island II is one of the video games featuring in the manga titled Cyber Boy, by Nagai Noriaki, Published by Coro Coro Comic and Shogakukan, from 1991 to 1993. Cover artwork for the packaging was illustrated by Shelley L. Hunter. Adventure Island II at MobyGames

Proustia is a genus of flowering plants in the gerbera tribe daisy family, native to South America and the West Indies. SpeciesProustia cuneifolia D. Don - Peru, Bolivia Proustia ilicifolia Hook. & Arn. - Argentina, Chile Proustia pyrifolia DC. - Chile Proustia vanillosma C. Wright - Cuba, Dominican Republic, Puerto Ricoformerly includedsee Acourtia Berylsimpsonia Lophopappus Vernonanthura Proustia crassinervis Urb. - Berylsimpsonia crassinervis B. L. Turner Proustia cuneata S. F. Blake - Lophopappus blakei Cabrera Proustia domingensis Spreng. Ex DC. - Vernonanthura buxifolia H. Rob. Proustia mexicana Lag. ex D. Don - Acourtia humboldtii B. L. Turner Proustia reticulata Lag. ex D. Don - Acourtia reticulata Reveal & R. M. King