TILTED PHOTOGRAPHS DEFINITIONS. Center for Photogrammetric Training Ferris State University
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1 TILTED PHOTOGRAPHS Center fr Phtgrammetric Training Ferris State University RCB DEFINITIONS Nadir pint (n) Grund nadir pint (N) Principal plane Principal line Azimuth (α) Tilt (t) Swing (s) Church angles (α, t, s) Tilted Phtgraphs 1
2 DEFINITIONS Iscenter (i) Axis f tilt L t t Camera lens axis +y' n i Axis f Tilt Vertical Line AUXILIARY COORDINATE SYSTEM In tilt-swing-azimuth system, auxiliary crdinate system required fr sme cmputatins Tw stages: Rtate abut principal pint thrugh an angle θ Translate alng principal line frm principal pint t nadir pint Origin at nadir pint, y axis cincides with the principal line, x axis is perpendicular t principal line at the nadir pint, clckwise 90 Tilted Phtgraphs
3 AUXILIARY COORDINATE SYSTEM AUXILIARY COORDINATE SYSTEM Rtatin angle θ s 180 Transfrmed crdinates after rtatin: x" x csθ ysin θ y" x sin θ + ycsθ Tilted Phtgraphs 3
4 AUXILIARY COORDINATE SYSTEM Translate rigin L Vertical Line t f Optical Axis Plane f Phtgraph n f tan t Final auxiliary crdinate values x' x csθ ysin θ y' x sin θ + y csθ + f tan t AUXILIARY COORDINATE SYSTEM Recgnize that sin θ -sin s and cs θ -cs s Then auxiliary crdinate system can be expressed as x' x css + ysin s y' x sin s ycss + f tan t Tilted Phtgraphs 4
5 AUXILIARY COORDINATE SYSTEM Nte: if rtatin angle defined as θ 180 s Then auxiliary crdinates becme x' x csθ + ysin θ y' x sin θ + y csθ + f tan t D nt mix frmulas frm the tw different appraches SCALE OF TILTED PHOTO In fllwing diagram wk is perpendicular t line Ln and is therefre a hrizntal line wp is perpendicular t the principal line and is als a hrizntal line Plane kwp is hrizntal plane Plane NWP is hrizntal plane Tilted Phtgraphs 5
6 SCALE OF TILTED PHOTO Scale between planes kwp & NWP fund by similar triangles kp NP Lk LN Lk H h Since Lk Ln kn f Then scale becmes f sec t S H y'sin t h sect y' sin t ( f ) y'sin t cs t H h Tilted Phtgraphs 6
7 GROUND COORDINATES FROM TILTED PHOTO On grund, the Y-axis lies principal plane In figure, wp x and WP X, then frm scale: ( f ) wp WP x' X y'sin t cs t H h X ( f ) cs t H h y'sin t x' GROUND COORDINATES FROM TILTED PHOTO Similarly, kw NW kw Y nw cs t Y y'cs t Y ( f ) y'sin t cs t H h Y ( f ) cs t H h y'sin t y'cs t Tilted Phtgraphs 7
8 EXAMPLE: SCALE ON A TILTED PHOTO Given the fllwing knwn values: t 3 7' x a " s 7 00' y a -1.00" H 7,850' x b -.866" f 6.000" y b -.010" h A 160' h B 70' Cmpute the scale at pints a and b EXAMPLE: SCALE ON A TILTED PHOTO Cmpute auxiliary crdinates Rtatin Angle: θ s Tilted Phtgraphs 8
9 EXAMPLE: SCALE ON A TILTED PHOTO Auxiliary crdinates: xa xa cs θ ya sin θ (+3.154" ) ( ) (-1.00" ) ( ) y a " xa sin θ + ya cs θ + f tan t (+3.154" ) ( ) + (-1.00" ) ( ) + (6.000" ) (0.0609) " xb xb cs θ y sin θ a +.109" y b xb sin θ + y -.43" b cs θ + f tan t EXAMPLE: SCALE ON A TILTED PHOTO Scale: SA f - y sin t cs t H - h 6.000" - (+3.549" ) ( ) ,850' 160' 1" 1,36' r 1 :15,918 SB 6.000" - (-.43") ( ) ,850' - 70' 1" r 1:13,896 1,158' Tilted Phtgraphs 9
10 EXAMPLE: GROUND COORDINATES FROM A TILTED PHOTO Cmpute crdinates f pints A and B X A 6.000" ,18 ' 7,850 ' -160 ' - (+3.549" ) ( ) (0.891" ) Y A 6.000" ,700 ' 7,850 ' -160 ' - (+3.549" ) ( ) (3.549" ) ( ) EXAMPLE: GROUND COORDINATES FROM A TILTED PHOTO X B 7,850' - 70' (.109") 6.000" ,44' Y B 6.000" ,81' 7,850' - 70' (-.43") ( ) (-.43") ( ) Tilted Phtgraphs 10
11 EXAMPLE: GROUND COORDINATES FROM A TILTED PHOTO Cmpute the grund distance between A and B AB ( - ) + ( - ) X B X A Y B ( 1,60' ) + ( 7,51' ) Y A 7,617' RELIEF DISPLACEMENT LN vertical line then planes LN A A and LNA are vertical planes Pints n, a and a lie in same vertical plane and n same line Pints n, b, and b lie in same line Therefre, relief displacement radial frm nadir pint Tilted Phtgraphs 11
12 RELIEF DISPLACEMENT Amunt functin f: Flying height Distance frm nadir pint t image pint Elevatin f grund pint Psitin f pint w.r.t. principal line and axis f tilt Relief displacement Will be less n upward side f pht Identical alng axis f tilt Greater n dwnward half f pht RELIEF DISPLACEMENT Amunt cmputed with sufficient accuracy using: d rh H r d r' h H h Distances r and r measured frm nadir pint Tilted Phtgraphs 1
13 TILT DISPLACEMENT L t/ t y* a n i t a" a' d t TILT DISPLACEMENT Amunt f displacement given as ( y *) dt ( f ) y * sin t y* is distance in y directin frm iscenter t the pint Apprximate frmula dt ( y * sin t) f Tilted Phtgraphs 13
14 TILT DISPLACEMENT Ex: Given the fllwing data, f 15.0 mm t 15 s cmpute the tilt displacement fr a pint given the measured pht crdinates x 7.08 mm y mm TILT DISPLACEMENT Slutin: the angle f rtatin is: θ ( ' ) Transfrm the y-crdinate t tilted pht crdinate system and cmpute translatin in y directin frm iscenter t give y* y* x sin θ + y csθ + f tan 7.08sin ( t ) ( 45 ) + ( 3.59) cs( 45 ) tan( 15' ) mm Tilt displacement is: dt y * y * 15mm sin t sin 15' ( f ) ( mm) ( mm) 1.8 mm Tilted Phtgraphs 14
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