Coordinate Systems L. M. Kalnins, March 2009

Transcription

1 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean in phsical space, but in geneal it simpl efes to what we might call vaiable-space, whee each dimension coesponds to one vaiable. A gaph of stock pices pobabl has vaiables of time and value, so we ae in time-value space, and ou coodinate sstem and an equations we might wite need to specif the time and value of each data point. How man paametes does ou coodinate sstem need to have? As man paametes as we have vaiables. This is elated to the idea of dependent and independent vectos. If we onl had two independent vectos, we couldn t get to all of the points in a thee dimensional object; likewise if we onl have two paametes, we can t specif data in tems of thee vaiables. Howeve, we saw that an thee independent vaiables will let us get to all of 3D space (will span 3-space, moe fomall). This also has an analog in coodinate space. The choice of paametes is not in geneal unique an two independent paametes will do, although thee ae onl a few standad choices. Diffeent Catesian Coodinate Sstems Catesian coodinate sstems ae familia, and we tend to pefe to oient them just so, with the -ais hoizontal and the -ais vetical. You should ecall fom Michaelmas tem, howeve, that it s possible to give a Catesian coodinate sstem an abita otation, and that an two ight-handed Catesian sstems can be elated to each othe b a otation. We geneall conside otations about a paticula ais, so that onl two of the thee coodinates change. The elationship between old and new coodinates can then be deived: cos sin sin cos = cos sin = sin + cos Figue 1: The elationship between coodinates in two diffeent Catesian coodinate sstems. Eecise 1: Use a sketch to deive the otation mati fo a coodinate sstem otation of π/3 clockwise about the -ais. Take seveal points in diffeent quadants and put them though the otation mati to find thei new coodinates. Use ou sketch to check if the answes seem easonable. 1

2 Pola Coodinates Catesian coodinates ae ve vesatile, but fo some applications including man cuves, otations, and comple numbes, it is simple to use a coodinate sstem based on the cicle. These ae pola coodinates, and ou two paametes ae, the adial distance between the point and the oigin, and, the angle between the point and the positive -ais. The simplest, most fundamental plots fo a Catesian coodinate sstem ae vetical lines lines of equal and hoizontal lines lines of equal. Fo the pola sstem, the fundamental plots ae cicles lines of equal and staight lines though the oigin lines of equal. = 2π/3 = 3 = π/2 = π/3 = 5π/6 = 2 = 1 = π/6 = π = 0 = 7π/6 = 11π/6 = 4π/3 = 3π/2 = 5π/3 Figue 2: The pola coodinate sstem. We can use geomet to elate Catesian coodinates to pola coodinates, just as we can elate diffeent Catesian coodinate sstems. You ve seen this with comple numbes, but it natuall also elates the eal Catesian plane to eal pola coodinates and can be used to tansfom equations in one sstem to the othe. = = actan( ) (pa attention to quadant) = cos = sin Figue 3: The elationship between pola and Catesian coodinates 2

3 Note that ou must pa attention to which quadant a point is in when conveting fom and to this is pecisel what ou ae doing when ou add ±π when figuing the agument fo a comple numbe with a negative eal pat (make sue ou see wh this is tue). In man applications, is alwas geate than o equal to zeo, but if is negative, instead of going outwads along the coect line of ou go backwads though the oigin and out in the opposite diection (along the line ±π). Fo eample, = 2, = π/2 is the same point as = 2, = π/2. Eecise 2: Plot the following pola equations b hand. Pola gaph pape is available at and ou can check ou answes using the gaphing function at kok/sg kok.html. = 4 = 5π/12 = /2 = 3 cos = 4 sin 4 Eecise 3: Epeiment with diffeent tpes of functions in pola coodinates using kok/sg kok.html, anothe compute pogamme, o a gaphing calculato. T vaious combinations of tigonometic functions as well as functions that ae common in Catesian coodinates such as polnomials. Figue out how to otate a function and how to enlage o shink it. Can ou do these easil in Catesian coodinates? What effects do simila manipulations of functions have in Catesian space? Note that the gaphing pogamme uses ˆ fo eponents, e.g. ˆ2 = 2. Eecise 4: Convet the following fomulae fom pola to Catesian coodinates o vice vesa and plot to check ou answes. It ma help to manipulate the equation to make the convesion simple; it should also be possible to aange ou answe into a fom that is familia (at least in Catesian coodinates). = 4 6 = 8 = 2 sin = 5/(3 cos ) Clindical Coodinates If ou ae familia with 3D Catesian coodinates and 2D pola coodinates, clindical coodinates ae a ve eas etension. The specif a adial distance and an angle in one plane, as fo pola coodinates, and a distance z pependicula to that plane. The elationship between clindical coodinates and Catesian ones is identical to that between pola and Catesian, with the addition of z = z. 3

4 z z = = actan( ) (pa attention to quadant) z = z = cos = sin z = z Figue 4: Clindical Coodinates Eecise 5: A child is plaing on a cokscew-shaped slide. The slide is 3 m tall, 1 m in diamete, and descends 0.75 m with each full tun. Find epessions fo the, z, and of the child s position in tems of time t if (a) the child goes though one tun of the cokscew in 2 seconds (b) the child appoaches the gound at 0.5 m/s (c) the child s velocit is 3 m/s. (These ae called paametic equations, when ou epess each vaiable in tems of some additional paamete such as time.) Which of these is the fastest, and which the slowest? What estictions should ou place on ou epessions to pevent the child fom magicall spiall though the ai o into the gound? Spheical Coodinates Like clindical coodinates, spheical coodinates can be viewed as a 3D etension of pola coodinates. In this case, the thid paamete is anothe angle, φ, measued fom the noth pole, and efes to the total distance of the point fom the oigin, not the distance in one plane. The eath s lines of latitude and longitude ae a familia sstem of spheical coodinates. Longitude is the, spanning 360 degees o 2π adians, latitude is the φ, spanning 180 degees o π adians, and we don t usuall bothe about the, since that is assumed to be adius of the eath. The othe diffeence is that spheical coodinate sstems in mathematics usuall use colatitude, measued fom the noth pole, athe than latitude, measued fom the equato. This means that φ spans 0 π athe than π/2 π/2. N.B.: Thee ae diffeent conventions on whethe is longitude and φ colatitude o vice vesa. I have used fo longitude to make the elationship between pola and spheical coodinate cleae, but it s a good idea to check which convention is being used befoe ou stat a poblem in spheical coodinates. 4

5 z φ = z 2 = actan φ = accos z = cos sin φ = sin sin φ z = cos φ Figue 5: Spheical Coodinates Eecise 6: Deive the elationships between Catesian and spheical coodinates given in Figue 5. Eecise 7: It is hopefull fail obvious what shape is given b constant. What shape is descibed b constant? Constant φ? What about b constant and constant φ? Constant and constant? Constant and constant φ? 5