The rotating Pulfrich effect derivation of equations

Transcription

1 The rotating Pulfrich effect erivation of equations RWD Nickalls, Department of Anaesthesia, Nottingham University Hospitals, City Hospital Campus, Nottingham, UK. 3 The rotating Pulfrich effect: erivation of equations The general case Determine m 1 m Determine m 1 + m Solve for x Solve for y The transition conition (φ/ε = 1) Viewing configurations Relate papers FROM: Nickalls RWD. Pulfrich geometry May 13, 2009

2 Chapter 3 The rotating Pulfrich effect: erivation of equations 1 Here we etail the erivation of the equations presente in the Appenix to the paper:- Nickalls RWD (1986). The rotating Pulfrich effect, an a new metho of etermining visual latency ifferences. Vision Research; 26, ( http: // ) 3.1 The general case Consier that the eyes (L, R; separation 2a) view an object P (the target) rotating clockwise about the center O with constant angular velocity ω. Let the eyes (L,R) be a istance from the center of rotation O such that the line LR is parallel to the y-axis (see Figure 3.1). If the target P is associate with angle θ, then let P lag behin P by angle φ (ue to a filter F in front of the right eye). The locus I of the apparent position is given by the intersection of the lines RP an LP. Let the lines RP an LP be given by where RP y = m 1 (x + ) a, (3.1) LP y = m 2 (x + ) + a, (3.2) m 1 = r sinθ + a r cosθ +, (3.3) r sin(θ φ) a m 2 = r cos(θ φ) +. (3.4) Solving these for x an y gives ( ) 2a x =, (3.5) m 1 m 2 ( ) m1 + m 2 y = a. (3.6) m 1 m

3 RWD Nickalls (2009) ROTATING PULFRICH EFFECT 6 Figure 3.1: Diagram (from above) showing the relative positions in the general case (see text) Determine m 1 m 2 m 1 m 2 = = ( ) ( ) r sinθ + a r sin(θ φ) a r cosθ + r cos(θ φ) + (r sinθ + a)r cos(θ φ) + } (r cosθ + )r sin(θ φ) a}. (r cosθ + )r cos(θ φ) + } Expaning an regrouping gives r 2 sinθ cos(θ φ) cosθ sin(θ φ)} + rsinθ sin(θ φ)} } m 1 m 2 = + arcosθ + cos(θ φ)} + 2a r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2. Since this simplifies to sinθ cos(θ φ) cosθ sin(θ φ) sinθ (θ φ)} sinφ m 1 m 2 = r2 sinφ + r2cos(θ φ/2)sin(φ/2)} + ar2cos(θ φ/2)cos(φ/2)} + 2a r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2, i.e., m 1 m 2 = r2 sinφ + 2r cos(θ φ/2)sin(φ/2) + 2ar cos(θ φ/2)cos(φ/2) + 2a r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2. (3.7)

4 RWD Nickalls (2009) ROTATING PULFRICH EFFECT Determine m 1 + m 2 m 1 + m 2 = = ( ) ( ) r sinθ + a r sin(θ φ) a + r cosθ + r cos(θ φ) + (r sinθ + a)r cos(θ φ) + } + (r cosθ + )r sin(θ φ) a}. (r cosθ + )r cos(θ φ) + } Expaning an regrouping gives r 2 sinθ cos(θ φ) + cosθ sin(θ φ)} + rsinθ + sin(θ φ)} } m 1 +m 2 = + arcos(θ φ) cosθ} r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2. Since sinθ cos(θ φ) + cosθ sin(θ φ) = sinθ + (θ φ)} = sin(2θ φ), this simplifies to r 2 sin(2θ φ) + r2sin(θ φ/2)cos(φ/2)} } m 1 +m 2 = + ar 2sin(θ φ/2)sin( φ/2)} r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2. Since sin( φ/2) = sin(φ/2) then an so we have 2sin(θ φ/2)sin( φ/2)} 2sin(θ φ/2)sin(φ/2), m 1 +m 2 = r2 sin(2θ φ) + r2sin(θ φ/2)cos(φ/2)} + ar2sin(θ φ/2)sin(φ/2)} r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2. Since sin(2θ φ) 2sin(θ φ/2)cos(θ φ/2) we can write 2r 2 sin(θ φ/2)cos(θ φ/2) + 2rsin(θ φ/2)cos(φ/2)} } m 1 +m 2 = + 2arsin(θ φ/2)sin(φ/2)} r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2, i.e., m 1 + m 2 = Solve for x From above we have 2r sin(θ φ/2)r cos(θ φ/2) + cos(φ/2) + asin(φ/2)} r 2 cosθ cos(θ φ) + rcosθ + cos(θ φ)} + 2. (3.8) ( ) 2a x = = 2a (m 1 m 2 ). m 1 m 2 m 1 m 2

5 RWD Nickalls (2009) ROTATING PULFRICH EFFECT 8 Substituting for (m 1 m 2 ), an after some manipulation, we eventually get x = 2ar2 cosθ cos(θ φ) r 2 sinφ + 2r cos(θ φ/2)acos(φ/2) sin(φ/2)} 2a + r 2. sinφ + 2r cos(θ φ/2)acos(φ/2) + sin(φ/2)} (3.9) Now we evelop 2ar 2 cosθ cos(θ φ) by expressing it in terms of cos(θ φ/2), using the following ientities: cosθ cos(θ φ) 1 cosφ + cos(2θ φ)}, 2 cos(2θ φ) 2cos 2 (θ φ/2) 1. Combining the two ientities 3.10 then gives 2ar 2 cosθ cos(θ φ) 2ar 2 cos 2 (θ φ/2) ar 2 + ar 2 cosφ, (3.10) which now allows us to substitute for cos(θ φ) in equation 3.9 an hence obtain x as a function of cos(θ φ/2) as follows } 2ar 2 cos 2 (θ φ/2) ar 2 + r 2 (acosφ sinφ) x = + 2r cos(θ φ/2)acos(φ/2) sin(φ/2)} 2a + r 2 sinφ + 2r cos(θ φ/2)acos(φ/2) + sin(φ/2)}. (3.11) Solve for y From above we have ( ) m1 + m 2 y = a. m 1 m 2 Substituting for (m 1 m 2 ) an (m 1 + m 2 ) we get ( ) 2r sin(θ φ/2)r cos(θ φ/2) + cos(φ/2) + asin(φ/2)} y = a r 2, sinφ + 2r cos(θ φ/2)sin(φ/2) + 2ar cos(θ φ/2)cos(φ/2) + 2a which after regrouping of terms gives y = 2ar sin(θ φ/2)r cos(θ φ/2) + cos(φ/2) + asin(φ/2)} 2a + r 2 sinφ + 2r cos(θ φ/2)acos(φ/2) + sin(φ/2)}. Finally, we expan r 2 sinφ 2r 2 sin(φ/2)cos(φ/2) an then ivie throughout by 2, to give y = ar sin(θ φ/2)r cos(θ φ/2) + cos(φ/2) + asin(φ/2)} a + r 2 sin(φ/2)cos(φ/2) + r cos(θ φ/2)acos(φ/2) + sin(φ/2)}. (3.12) Equations 3.11 & 3.12 are therefore the parametric equations of the path I, as θ increases from 0 to 360eg. When P rotates clockwise θ is consiere to be +ve; with anticlockwise rotation θ is consiere to be ve. Now equation 3.11 is quaratic in cos(θ φ/2) an so θ can be eliminate to give the Cartesian equation, by substituting the roots of equation 3.11 for cos(θ φ/2) into equation In the general case this gives rise to a complicate higher curve which is symmetric about the x-axis, as shown in figure 3.2 (from Nickalls, 1986). The two curves shown in figure 3.2 represent equations 3.11 an 3.12 for a range of values of the parameter φ/ε, with P rotating clockwise. When φ/ε < 1 then the Pulfrich construction (P x ) is also clockwise. When φ/ε > 1 then the locus is anti-clockwise. The special transition case, when φ/ε = 1, is iscusse in the next section.

6 RWD Nickalls (2009) ROTATING PULFRICH EFFECT 9 Figure 3.2: These graphs show the theoretical curves (Equations 3.11 an 3.12) inicating the preicte apparent paths associate with ifferent latency ifferences (i.e., with ifferent filters) while fixating a vertical ro, rotating clockwise in a horizontal circle, from a istance of 200 cm to the left of the centre, as in the arrangement shown in Fig. 3.1 (2a = 6 6 cm; ω = 45 1 rpm; ro is 20 cm from the centre; axes are in cm.) The parameter (latency ratio) is the latency ifference expresse as a multiple of that require to see transition uner these circumstances (7 msec). The +ve sign inicates that the filter is in front of the right eye; ve sign inicates the filter is in front of the left eye. Arrows inicate the irection of apparent rotation. Thick circle (latency ratio = 0) inicates the actual path of the rotating ro ( t = 0). The thick arc (latency ratio = +1) inicates the apparent path at transition ( t = 7 msec). Top: Latency ratios from 0 5 to +1, clockwise rotation. Bottom: Latency ratios from +1 to +2 5, anticlockwise rotation.

7 RWD Nickalls (2009) ROTATING PULFRICH EFFECT The transition conition (φ/ε = 1) This is the special transition case which is associate with the conition φ = ε. In this case the locus I is an arc of the circle LRO as shown in the following figure. Figure 3.3: Diagram (from above) showing the relative positions in the general case (see text). At transition I passes through the center O an so Substituting equation 3.13 into equation 3.11 gives an x = tan(φ/2) = a/ (3.13) (r 2 /)(a )sin 2 (θ φ/2) a r 2 + 2r a cos(θ φ/2), (3.14) (r/)(a )r cos(θ φ/2) + } a y = ±sin(θ φ/2) a r 2 + 2r. (3.15) a cos(θ φ/2) Eliminating θ from equations 3.14 an 3.15 gives ( a y ) = x + x, (3.16) which represents a circle passing through L,R an the origin O, where R = (a )/(2) as shown in the figure above. The equation can therefore be expresse in terms of R as follows. y 2 = x(2r + x).

8 RWD Nickalls (2009) ROTATING PULFRICH EFFECT 11 It is important to note that the path at transition is not necessarily the whole circle as, epening on the conitions, only part of the arc is mathematically real. The limits are foun by solving equation 3.15 for θ as follows. Conveniently equation 3.15 can be turne into a quaratic by using the ientity sin 2 (θ φ/2) 1 cos 2 (θ φ/2) which generates (r 2 /)(a )cos 2 (θ φ/2) 2xr } a cos(θ φ/2) x(a r 2 ) (r 2 /)(a = 0. ) Diviing throughout by the coefficient of cos 2 (θ φ/2) an rearranging we get ( ) 2x cos 2 (θ φ/2) r cos(θ φ/2) = x a r 2 + x a We can now use the metho of completing the square to solve the quaratic by aing the term (x/(r (a )) 2 to both sies as follows. which gives ( cos(θ φ/2) an so we have ( cos(θ φ/2) cos(θ φ/2) = x r a x ) 2 = ( r a x r a ) 2 + x r 2 + x a , ) 2 ( )( ) x x = r a , (x )( ) x x r a 2 + ± 2 r a (3.17) Now since from the geometry we have R 2 = a 2 + ( R) 2 then 2R = a2 + 2, an so finally we have (x ) ( x ) cos(θ φ/2) = x 2Rr 2 ± r R + 1. (3.18) It follows therefore that for equation 3.15 to yiel real solutions we nee the contents of each of the square-root terms to be 0, that is we nee /R 0 an ( ) x ( x ) r R + 1 0, for which we nee either ( ) x r or ( ) x r AND AND ( x 2R + 1 ) 0 ( x 2R + 1 ) 0,

9 RWD Nickalls (2009) ROTATING PULFRICH EFFECT 12 where, r, an R are of course all negative owing to our chosen coorinate system (centere at the center of rotation) an configuration (viewing from the left). However, x cannot be less than 2R (since the locus is the arc of the circle raius R), an so only the first conition is relevant for a real locus, i.e x r2 AND which is consistent with the single conition x 2R, Viewing configurations x r2. (3.19) There are three configurations which may now be consiere, epening on whether the points L,R are outsie the circle, on the circle, or insie the circle. In practice however, since the observer is outsie the circle of rotation, we nee only consier the case for which L,R lie outsie the circle ( a > r 2 ), i.e., r 2 ( a 2 > + 2 ), for which the conition x r 2 / is sufficient, as this is always consistent with x (a )/, an hence we have r 2 x 0. (3.20) 3.3 Relate papers The following relate papers aress the visual an/or geometric aspects of the rotating Pulfrich effect. They are all available from Nakamizo S, Nawae H, an Nickalls RWD (1998). Precision of the rotating Pulfrich technique for etermining visual latency ifference is significantly greater if viewing istance is varie, than if angular velocity is varie. Perception; 27 (Supplement), 97. (Abstracts of the 21st European Conference on Visual Perception; Oxfor, Englan; August 1998). com/abstract.cgi?i=v Nakamizo S, Nickalls RWD an Nawae H (2004). Visual latency ifference etermine by two rotating Pulfrich techniques. Swiss Journal of Psychology, 63, Nickalls RWD (1986). A line-an-conic theorem having a visual correlate. Mathematical Gazette; 70 (March), 27 29, (JSTOR). ick/papers/maths/lineanconic1986.pf Nickalls RWD (1996). The influence of target angular velocity on visual latency ifference etermine using the rotating Pulfrich effect. Vision Research; 36,

10 RWD Nickalls (2009) ROTATING PULFRICH EFFECT 13 Nickalls RWD (2000). A conic theorem generalise: irecte angles an applications. Mathematical Gazette; 84 (July), , (JSTOR). nickalls.org/ick/papers/maths/conicthm2000.pf Nickalls RWD, Kazachkov AR, Vasylevska Yu.V an Kalinin VV (2002). Motional visual illusions on-line. Proceeings of the 2002 International Conference on Information an Communication Technologies in Eucation (ICTE2002); Baajoz Spain; November 13 16, 2002; p (ISBN: Coleccion ). pf