ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Progress Report A METHOD FOR CORRECTING AERIAL PHOTOGRAPHS FOR IMAGE DISPLACEMENT CAUSED BY SUPERSONIC SHOCK WAVES WHEN THE SHOCK-WAVE CHARACTERISTICS ARE KNOWN W. H.,\Ball E Young: J, M. Vukovich ERI Project 2426 RAYTHEON MANUFACTURING COMPANY MAYNARD LABORATORY MAYNARD, MASSACHUSETTS May 1958

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The University of Michigan ~ Engineering Research Institute ABSTRACT In supersonic flight, the images in an aerial metric camera with optical axis vertical, or nearly so, will have displacements varying in-magnitude depending on the angular positions of the ground objects with respect to the optical axis of the camera and the configuration of the shock wave, In this discussion a 120~ cone* is assumed as the shock-wave pattern and the angular field of the camera is taken as 90 degrees. The effect of refraction due to boundary layer is mentioned in the text and reference is given to work done on this subject. See schlieren photographs in Reference lo ii

The University of Michigan ~ Engineering Research Institute INTRODUCTION There are two main effects in supersonic flights that result in refractio of an image ray while taking aerial photographs. One of these is caused by the boundary layer.which is the transition zone where a change in wall to freestream temperature occurso This temperature differential changes the density of the air across the boundary layer and hence causes the incident image ray to be refracted. Graphical representations of this deviation plotting 5/tangi vs. Tw/To for altitudes from 0 to 75,000 feet, where 8 is the deviation in seconds of arc,. ii is the angle of incidence of the light ray, Tw is the wall temperature, and T, is the free-stream temperature, have been presented in a report by Mo P. Moyle and Ro. Eo Culleno These graphs, or graphs similar to them, could easily be used to correct the aerial photogaph for deviations caused by the boundary layer, From the shape of the curves, -it:would be a simple matter to program an analytic. correction procedure. The other of these effects is caused by the bending of an image ray when passing through the shock wave. In this discussion it has been assumed that information on the shock-wave configuration, as well as information on how a light ray is bent while passing at various angles of incidence through a shock wave, could be made available to us. DISCUSSION From the report by Moyle and Cullen it seemed reasonable to assume that the shock-wave pattern could be represented by a 1200 cone. To begin with, the equations for the family of cones representing varying field angles for a vertical camera were computed. We let Q equal one-half the field angle and computed the family of camera or image cones for 50 increments of, G. Next the intersection of the 120~ shock-wave cone and the camera cone with G - 45~ was computed (see Fig. 1). The initial point of intersection of these two cones has coordinates (0,-O.634d,-0o634d), where d is the distance from the apex of the shock-wave cone to the camera (see Fig. 2). Then the family of.circles corresponding to the locus of all points where an incident image ray intersects the shock wave at some angle a was computed. This family was computed for 5~ increments of a. In Fig. 1 the circles for a = 75~ and for a = 300 are displayed. These circles have equations of the form x2 + z2 = r2 lying in a plane associated with a particular value of y. For example, the circle corresponding to a = 75~ is represented by x2 + z2 = (0o,634d)2 = 0O4020d2 lyin in the plane y = -0o,634do Then the intersection of each of these equal a

The University of Michigan ~ Engineering Research Institute circles with each of the family of image cones was computed. The images on the photographic plane of these intersections were now desired, To obtain these it is necessary to project back through the lens, but-since these intersections do not lie in a plane, a further correction has to be made (see Figso 3'and 4), Note that, after this correction, the images of the intersections, written in X,Y coordinates, are functions of f, the focal length of the camera, and are independent of d. Figure 5 is the final plotting of a photographic plate with the images of the intersections for various values of c6. These curves are hyberbolic and exhibit symmetry about the X and Y axes. As an example of the method of correction, any point on the line o = 50~ would have been deviated by some specified amount, whatever the deviation a light ray passing through a shock wave at a 50,0 angle of incidence was found to beo Since a vertical camera was assumed, the correction is radial to the principal point. Na urally, interpolation would be necessary for thoe image points on the photo which did not fall on a line corresponding to a tabulated value of ca. Note that the system is not symmetric with respect to the camera angle H. Hence it is thought that a graphical correction would be simpler to obtain than an analytic correction, An immediate method of correction would be to produce a copy of Fig. 5 the size of a 9-1/2-by-9-1/2-inch photo and to superimpose this copy on a photo to see where the images of the control points fall, and correct accordingly. At any rate, further efforts will be made to obtain an analytic correction method. It seems advisable to discuss the various assumptions that were made as well as the changes that would occur if the assumptions were not valido Io A 1200 shock-wave cone was assumed. In applying a correction of this type, it would be necessary to know the actual shock-wave configuration for the plane in useo The shock wave may not be strictly conical, in which case -this method-would have to be modified or discarded depending on how great the deviation from a conical shape was. The cone may have an apex angle not equal to 120~, but this could easily be corrected for by changing the equation for the shock-wave cone in an appropriate mannero There also may be a series of shock waves of this type, in which case a repeated application of this method could be applied. 2, A vertical camera, that is, zero tilt, was assumed. If there is a departure from verticality, two things occur, First, symmetry about the principal point is destroyed. Second, corrections would be radial to the nadir point for tilted photographs and not to the principal point as when zero tilt is the case. In low-tilt photographs the magnitude of the change this introduces is probably such that it could be ignored. 3. The apex of the shock-wave cone and the camera were assumed to lie on the longitudinal axis of the plane. If the camera were mounted outside the longitudinal axis, symmetry would occur about both the image of the line connecting the camera and the apex of the shock-wave cone, and about a line perpendicular to this line through the principal point of the photo if -the camera 2.

The University of Michigan ~ Engineering Research Institute - were not tilted. With an offset, tilted camera, there would be a combination of these effects. One other point remains to be mentioned and it is something that would have to be investigated further. For a particular velocity of the plane, there is a particular shock-wave configuration and a particular shock-wave strength. For a different velocity, the shock-wave configuration probably remains relatively the same but the wave strength changeso Thus, the manner in which a light ray is bent while passing at various angles through a shock wave is probably a function of the wave strength and hence of the Mach number.

The University of Michigan ~ Engineering Research Institute Shock wove cone (O,- d, O) Equal a circle 30- - 7a 6 <"~ /1 / \ I\ ~'% -450 camera image cone g= 60-:45\ Fig. 1. Intersection of 1200 shock-wave cone with family of camera coneso

The University of Michigan ~ Engineering Research Institute (O,-d,O) r/ d(- 0r0/0) 60 y 908 tan d - r/ tan = r r r tan g = d r/ y = r tan g r = tan G + tan 30~ Fig. 2. Radius of equal ax circles.

The University of Michigan ~ Engineering Research Institute The family of image cones is given by (cot2 G)(x2 + y2) _ z2 = 0 450 cone: lo0000 (x2 + y2) _ z2 = 0 40~ cone: 1 4204 (x2 + y2) -z2 0 350 cone: 220395 (x2 + y2) _ z2 = 30~ cone: 3~0000 (x2 + y2) _ z2 0 25~ cone: 4-5989 (xA + y2) -_ z = 0 200 cone: 7. 5488 (x2 + y2) _ z = 15~ cone: 13.9286 (x2 + y2) _ Z2 = 0 100 cone:: 32.1636 (x2 + y2) -2 = o 5~ cone: 130.6449 (X2 + y2) _ z2 = o

The University of Michigan * Engineering Research Institute The equations for the equal a circles are as follows: r Y x + z2 75 o 634d -0o 634d 0 4020d2 70 ~ 0706d -0o 592d 0o4984d2 65~ o 0783d -0 548d 0 o 6131d2 60~ o 866d -0o 500d 0O7500d2 550 o 958d -o.447d 0 9178da 50~ lo062d -0 387d 1.1278d2 45 - 1.183d -00317d 103995d2 40 o 10327d -0O 234d 1o7609d2 35 ~ 1504d -0.132d 2.2620d2 30 ~ 1732d 0oOOOd 3.0000d2 250 2041d 0 179d 4.1657d2 200 2 494d 0 440d 6o2200d2 15 ~ 3232d 0o866d 10o.4458d2 10~ 4.686d 1o706d 21o9586d2 5~ 9.006d 4 o200d 81.1080d2

The University of Michigan * Engineering Research Institute The results of the computation for the initial intersection curves between the equal a- circles and the family of image cones are as follows: cone x y z 750 45 O o O0d -o634d -0 634d 70 45 0 +0 o272d -0. 592d -0.652d 70~ 40~ o O00Od -0o592d -0.706d 650 45 +o ~0395d -0.548d -0.676d 650 40 ~ +0.278d -0o 548d -0.732d 65 35 0.000 d -0o. 548d -0.783d 60~ 45 +0O500d -0.500d -0-707d 60~ 400 +0.404d -0o500d -0.766d 60 35 0 ~0.281d -04500d -0.819d 60~ 530 0.000d -0.500d -0.866d 550 45 ~ +~0.599d -0.447d -0.748d 550 40~ +0o512d -0.447d -0.810d 550 35 +o0.410d -0.o447d -0.866d 55 ~ 30 ~ +~0282d -0.447d -0o916d 550 250 0.000 d -0.447d -0.958d 50 450 +0.699d -0o387d -0o 799d 500 40 +0.615d -0. 387d -0.866d 500 355 +0.520d -0.387d -0.926d 50 ~ 30~ +0411d -o.387d -0o 979d 500 250 +0 o280d -0.387d -1.024d 50~ 20 0.000 oOOOd -0O387d -1.062d 450 45 0 +o806d -0o 317d -0o 866d 45 ~ 40 o~072Ad -0o.317d -0O 938d 45 355 +o0.627d -05317d -1.003d 450 30~ + 0.524d -0o317d -1o061d 450 25 +0.409d -0.317d -1.110d 45~ 200 +0 o274d -0.5317d -1.151d 45 15 0.000d -0.5317d -1.183d

The University of Michigan * Engineering Research Institute o~ cone x y z 40 ~ 45 ~ +0924d -0o234d -0o953d 40 40 +0o.834d -0.234d -1.032d 40o 3550 +0o37d -0.234d - 1104d 40o 30~ +0o632d -0O234d -1.167d 40~ 25~ +~0o519d 0-0.234d -1 221d 40~ 200 +0o 397d -0 234d -1.266d 40~ 15 +0 266d -0o234d -1.300d 40~ 10~ 0000d -0.234d -1.327d 350 450 +1.059d -0.132d -i.068d 350 4o0 o +0o961d -0o.132d -1.157d 35 350 o00856d -0.132d -1.237d 355 300 +00743d -0.132d -1o308d 35 250 +0,o624d -0.132d -1.368d 35 20 +0o.499d -0.132d -1.419d 35~ 15~ +0,352d -0.132d -1 462d 35 100 +0,.226d -0.132d -1.487d 35 5~ o000ooo -0132a -1.o504d 30~ 45~ +1 o.225d O o000d -1 o225 30 40 +o 113d o 0.000d -1.327d 30~ 355 0o 993d O oOOOd -1 o419d 30 30 ~0993 o.oooa -1.600d 30 3150 +o.686a o.oooad -1.675oo6 30 25 +01o732d o0OOO. -1.570d 300 20~ +0.592d O 0OOO -1o6286 30 15~ +0.448d 0.000d -1o673d 30 0 ~~ 0. 301d O o OOOd -1.o706d 30 5 ~ +00151d 000 o OOOd -1o725d 30o O ~ O.000 d 0,00 o OOOd -1 b732d 250 450 + 1.438d 0o179d -1.449d 25 40~ +1o.305d 0.179d -1.570d 250 35 +o1.161d 0.179d -1.678d 25~ 30~ loO09d 0O179d -1.774d 25~ 250 +0o847d ol179d -1.857d 25 20 o +o 677d 0 o 179d -1o925d 25 ~ 15 +0o499d 0.179d -1o979d 250 100 ~0.307d O 0179d -2. 018d 250 50 0oOOOd 0.179d -2.041d

The University of Michigan * Engineering Research Institute cl cone x y z 200 45 ~ +1736d 00440d — 1.791da 200 40 ~ +1567d 0.440d -1 940d 20 355 +1 384d 0o 440d -2 074d 200 30 +1o 187d 0o440d -2.193d 20 25 +0 976d 0o440d -2 295d 200 20~ +0 o746d 0.440d -2 380d 20 15 ~ +0o486d 0o.440d -2o446d 20~ 100~:0o OOOd o.440d -2 2494d 150 45 o +2 ~ 202d o 866d -2 366d 150 40o +1 969d 0.886d -2.563d 150 350 +1o 713d 0.866d -2 741d 150 3~0 ~+1431d 0o866d -2.898d 150 250~ ~+1118d 0o866d -3.033d 150 20~ _+0749d 0 o866d -3 144d 15~ 15~ 0 o OO.d o 866d -3.232d 10~ 45 0 +3o 086d 1 706d -3o 526d 10 ~ 40~o +2.714d 1.706d -3.820d 10 o 3~550 ~2o296d 1.706d -4o 085d 100 300 ~1f818d 1.706d -4 n319d 100 25 +1 o237d 1706d -4.520d 10~ 200 0 000d 1o706d -4o686d 5 45 ~5.633d 4.200d -7.027d 5~ 40~ +4.812d 4o200d -7.612d 5 35 ~ +3 o853d 4 200d -8 140d 50 30~ +2o655d 4.200d -8o606d 10

The University of Michigan ~ Engineering Research Institute (0,,, 0) X / \Corrected intersection ~ ~ ffcurve /Initial intersection curve Fig. 3. Projection through camera lens of intersection curves into focal plane, 11

The University of Michigan ~ Engineering Research Institute (0,~0,0) z=-0.634d AR Az 1/2 R AR R = (X2 + y2) tan =-= -A z Az AR = Az tan y x f Ax -- AR = x - x X =x R o.634d Ay- Y AR y' = y Ay - 0.654d Ry - Fig. 4. Z-difference correction to points of intersection. 12

The University of Michigan * Engineering Research Institute The results of the computation for the corrected intersection curves are as follows: c~ X Y 75 o OOO0f -o 0f 70 +0o.416f -0 909f 70~ O o OOOf -0o839f 65 +o0584f -0o811f 65~ +05380f -0.749f 650 ooo000f -0o700f 600 ~0o707f -0 707f 60~ +0 o527f -o o653f 6o0 +0.342f -o061of 60 0. o OOf -0o 577f 55~ +0 o801f -0.598f 550~ +0o632f -0,552f 550 +0.473f -0o.516f 550 +~0o308f -0o.487f 550 0o000f -00467f 50~ +0o875f -0.484f 50~ +0 710f -07446f 500~ +0562f -0o418f 500 ~o0420f -0 396f 50~ +00273f -0.379f 50~ OO000f -0o364f 450~ +0o931f -0 o366f 45~ +0o770f -0 338f 45 +o0.625f -0o315f 450~ -+0494f -0. 300f 45~ +0 369f -0o285f 450~ +0 238f -0o 276f 450 0o000f -0o268f 13

The University of Michigan * Engineering Research Institute X Y 40~ +0~ 970f -0.246f 40~ +0o808f -0.227f 40~ +~.667f -0 o211f 40 ~ +0 541f -0o200f 40o~ +0o424f -00191f 40~ +o0314f -o.185f 40~ +0.205f -o.18of 40~ 0 oOOOf -0o177f 35~ +0 0992f -0.123f 350 + o831f -o.114f 35~ +0o692f -Oo107f 35 +0o 568f -0.101f 35~ +0 o456f.-0o096f 355~ +0352f -0oO93f 350~ +0241f -o0o9Of 50 ~O.l55f -o o88f 35 ~ +o 153 - o 088~ 35 ~ 0.000f -o o88f 30 ~1o 000f O.OOOf The rest of the a = 300 all lie on the x axis. 250 +0 992f 0o125f 25~ +00831f O114f 250~ +0o692f 0 107f 25 ~ +ol568f 0,nOlf 250 +0.456f o.o096f 25~ +00352f 0o093f 250 ~0 252f 0o.090f 25~ +O 0153f o. o88f 250 0o000f o.o88f 20~ +0 970f 0.246f 200 +0.808f 0o227f 20"~ +0667f 00213f 20~ +0.541f 0o200f 20" o0426f 0 o192f 20" +o0314f o.085f 20~ +o~099f0 o 180f 20 o 0000f 0o177f 14

The University of Michigan * Engineering Research Institute x Y 15~ +0o931f 0 o364f 150~ ~o768f 0o338f 15~ +0~o625f 0o315f 150~ +0o494f 0o298f 15 +0 o369f 0.285fi 150~ +0o238f 0o276f 150~ O00f 0.268f 10~ o o0875f 0o484f 10~ +o0710f oo446f 100 +O 562f o.418f 100 +~o421f 0o394f 10~ +o0273f 0o377f 100 0000f o0364f 5 +0 o801f 0.596f 50~ +0o632f 0o552f 50 +0 o473f o0516f 5~ +0.309f o.489f ~5~ O o0OOOf o.465f

The University of Michigan Engineering Research Institute 120.....~~............~~ - a =0z _t"a ~ ~ ~ ~ ~ ~ ~ = loot __ (extrapolated) ~'~~~ I 00 - "'".-.,.......~.~'~, ~~~a.= 5~ 80 a -10 60 a —_5 _.._40.. 200' L;L z~ L | a =25' 20 -,I,,1, I' I-,. I.I 0I a~,l I I I, I,I....0 -120 -100 -80 -60 -40 -20 20 40 60 80 100 120 a =350 -20 a=400 -40 a =4511 - 60 a =-500 - 80......~.:'~~d 55~ -100 a =60~ -120 a=650 a a-70~ -140 — an75~ Computed for f = 150 mm. Fig. 5. Plot of equal a curves of intersection~

The University of Michigan * Engineering Research Institute REFERENCE 1o Moyle, M. P,, and Cullen, R. E., Anti-Icing and Anti-Frosting of Aerial Photographic Windows, Univ. of Mich. Eng. Res Inst. Report 2197-25-F, Ann Arbor, October, 1955. BIBLIOGRAPHY 1. Moyle, M. P., and Cullen, R. E., Refraction Errors in Aerial Photography at High Flight Speeds, Univo of Mich. Eng. Res, Inst. Report 2197-14-P, Ann Arbor, May, 1955. 2. Moyle, M. P,o and Cullen, R. E., Refraction Errors in Aerial Photography at High Flight Speeds, Univ. of Mich. Engo Res, Inst. Report 2197-20-P, Ann Arbor, May, 1955. 3- Moyle, M. P., Jackson, Po L.., Dabora, E. K., Sherman, P., and Cullen, R.E.,o Experimental Evaluation of the RF-101 Forward Oblique Window Under Stress and Effect of Air Density Changes on Aerial Photography, Univ. of Mich Eng. Res. Inst. Report 2508-1-F, Ann Arbor, April, 1957. 17

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