Three-dimensional graphs

Transcription

1 Thee-dimensional gaphs The Gaphing Calculato can gaph seveal foms of thee-dimensional equations. Gaph Eplicit function suface plot Implicit suface Eample cos Paametic cuve cos 4πt sin 4πt 2t Paametic suface 2u - 1 2v - 1 cos 4uv Vecto field Point 1 2 n 3D Coodinate tansfomation ' ' 2 ' Gaphs involving comple numbes Comple paametic cuve + i e 2πi Comple suface Re (sin ) Eplicit function suface plots The calculato will daw sufaces fom equations in an of the following foms: ƒ(, ) ƒ(,) ƒ(,) ƒ(, ) ƒ(, ) ƒ(,) ƒ(, ) ƒ(, ) ƒ(,)

2 The suface is dawn fo the anges of the independent coodinates shown on the ight side of the math pane. You can change these anges, if desied. Eamples: cos ep -( ) sech 2 + sech 2 cos + sin cos ( + ) 2 sin 2 cos 2 4 cos sin + sin 3 The calculato will also ecognie functions of the fom ƒ(,,,), whee is a clindical coodinate. In this case, and ae not independent coodinates. Eample: sin sin 3 + cos Fo sufaces of the fom ƒ(, ), ecept fo the cases ƒ( ) and ƒ( ), ou can omit " " and tpe just the epession on the ight hand side. You can also omit " " fo sufaces of the fom ƒ(,) and of the fom ƒ(,,,). Implicit sufaces The calculato will daw a suface defined b an implicit elation in coodinates, and. Eamples: sin 2 + sin 2 + sin sin Even though this equation can be witten in the eplicit fom sin + 2, the calculato will conside it an implicit elation.

3 Implicit sufaces ae dawn onl in the,, and anges shown at the ight side of the math pane. Neithe the etended colo menu no tetue maps ae available fo implicit sufaces. Paametic cuves A paametic cuve is a locus of points whose coodinates ae given as functions of a paamete t. You can wite equations in the following foms: Geneal Fom Eample Daw the cuve taced b the equations f(t ), g(t ), and h(t ) f(t) g(t) cos 2πt sin 2πt h(t) t Daw the cuve taced b the equations f(t ), g(t ), and h(t ) f(t) g(t) 2 cos 2πt sin 2πt h(t) 2t(2 - t 2 ) Daw the cuve taced b the equations f(t ), g(t ), and h(t ) f(t) g(t) 3t 5t h(t) 7t The ange of t fo which the cuve is dawn is shown at the ight side of the math pane. You can change this ange, if desied, b editing it manuall. The default ange is 0 to 1. Paametic sufaces The set of points whose coodinates ae given as functions of two paametes u and v make a paametic suface. You can wite equations in the following foms: Geneal Fom Eample Daw the suface defined b the equations f(u,v ), g(u,v ), and h(u,v ) f(u,v) g(u,v) (1 + v sin 2πu) sin 2πv v cos 2πu h(u,v) (1 + v sin 2πu) cos 2πv

4 Daw the suface defined b the equations f(u,v ), g(u,v ), and h(u,v ) f(u,v) g(u,v) 2 + cos 2πv 2πu h(u,v) sin 2πv Daw the suface defined b the equations f(u,v ), g(u,v ), and h(u,v ) f(u,v) g(u,v) h(u,v) 1 + cos πv + sin πu 2πu(v v + u v ) πu The anges of u and v fo which the suface is dawn appea on the ight side of the math pane. You can can change these anges. The default anges of u and v ae both 0 to 1. Vecto fields To daw a thee-dimensional vecto field, define the,, and components as functions of the coodinates,, and b witing an equation in the fom: f(,,) g(,,) h(,,) Eample: Daw the vecto field V(,, ) f(,, ) i + g(,, ) j + h(,, ) k whee i, j, and k ae the unit vectos in the,, and diections espectivel 1 - Fo eve point in space, this defines a vecto with component one, component equal to minus the coodinate of the point, and component equal to the coodinate of the point. The calculato evaluates the vecto field on a egula lattice of points filling the gaph egion. Vectos ae dawn with thei bases at these points, and thei lengths ae scaled so that the longest vecto fits between adjacent lattice points. Vectos with infinite o undefined components ae not dawn. It can be visuall confusing when 3D vectos ae dawn at eve point in a cubic lattice. You can limit dawing to onl vectos on a suface. An equation fo an eplicit function

5 You can limit dawing to onl vectos on a suface. An equation fo an eplicit function suface plot, implicit suface, o paametic suface, can be witten afte the equation fo a vecto field, sepaated b a comma. The suface itself won't be displaed, but the vecto field will onl be dawn at points on the suface. 0.1, 3 Shows an outwad pointing vecto field on the suface of a sphee. Points The calculato can gaph a single point in space b specifing its,, and coodinates. You specif coodinates b witing an equation in the fom: # # # Eample: D coodinate tansfomations The calculato can visualie 2D and 3D coodinate tansfomations. It daws side-b-side gaphs with the untansfomed, unpimed coodinates on the left and the tansfomed, pimed coodinates on the ight. Eample: Daw a sphee of adius one on the left and scale it b a facto of two on the ight. Ente and gaph the following two equations togethe: ' ' '

6 Comple paametic cuves The calculato can plot comple functions of a eal paamete as a cuve in thee dimensions. You can specif the cuve b witing an equation in the fom: + i ƒ( ) which is almost equivalent to: Re ƒ(t ) Im ƒ(t ) t ecept that the ange of is detemined b the view. Eample: + i e 2πi Comple sufaces The calculato can plot comple functions as sufaces in thee dimensions b specifing a eal pat using Re, lm, ag, o. You can specif the cuve b witing an equation in the fom: ƒ( ) whee ƒ( ) evaluates to a eal numbe. This is equivalent to: ƒ( + i ) Eamples: Re (sin ) Im (sin ) sin Re (sin sin πn) + Im (sin sin πn)