The study of animal movement is an important aspect of functional morphological research. The three-dimensional movements of (parts of) animals are usually recorded on two-dimensional film frames. For a quantitative analysis, the real movements should be reconstructed from their projections. If movements occur in one plane, their projection is distorted only if this plane is not parallel to the film plane. Provided that the parallel orientation of the movement with respect to the film plane is checked accurately, a two-dimensional method of analysis (ignoring projection errors) can be justified for quantitative analysis of planar movements.

Keywords:
cinematography, projection errors, vector method

Films of movements of skeletal elements of the fish head have generally been analyzed with the two-dimensional method (e.g. Sibbing, 1982; Hoogenboezem et al. 1990; Westneat, 1990; Claes and de Vree, 1991), which is justifiable for planar movements. Unfortunately, the movements of the head bones of fish are often strongly non-planar, e.g. the movement of the pharyngeal jaws and the gill arches. The two-dimensional method is inappropriate for studying such complex movements (Sibbing, 1982; Hoogenboezem et al. 1990). For a qualitative description of movement patterns, the conditions for the use of the two-dimensional method may be somewhat relaxed.

When two (or more) views of a movement are recorded simultaneously, the three-dimensional movements can readily be reconstructed using two two-dimensional images (e.g. Zarnack, 1972; Nachtigall, 1983; van Leeuwen, 1984; Drost and van den Boogaart, 1986). However, because of technical (and budget) limitations, simultaneous views of a movement cannot always be shot. In this paper, a method is presented for reconstructing the three-dimensional orientation and rotational movement of structures using single-view films and for calculating rotation in an object-bound frame. Ellington (1984) presented a similar method for determining three-dimensional wing movements from single-view films of flying insects. Ellington’s method is based upon the bilateral symmetry of the wing movements. The present method does not depend on symmetry and can be applied to a variety of kinematic investigations. It eliminates a systematic error: the projection error. The measuring error is not discussed; it is the same in the two-dimensional and three-dimensional method of analysis.

The three-dimensional method of analysis can only be applied when the following general requirements are fulfilled. (1) The magnification of the projection of the object should be known (e.g. with the aid of a scale bar parallel to the film plane). (2) At least two markers should always be visible in each structure to be analyzed. These markers should be as far apart as possible in the direction of the movement under study. Markers may be conspicuous and well-defined anatomical points or artificial points (e.g. surgically implanted pieces of platinum, which are commonly used in X-ray cinematography). (3) The distance between the markers in each structure should be known accurately in each frame (a constant distance is most convenient). This can be determined in the anaesthetized animal. (4) One should know whether the structures are pointing ‘up’ or ‘down’ with respect to the film plane. This cannot always be determined from the film frames. The easiest way to solve this problem is to make sure that the angle between each structure and the film plane stays well within the range 0–180°; in other words, to make sure that the structure is either pointing ‘up’ or ‘down’ during the entire film sequence. For rotations with very large amplitude, this may be impossible. In general, direct or video-recorded observation of the animal during filming is enough to judge whether a particular film sequence is suitable for analysis (e.g. when the animal turns on its back the sequence may not be suitable).

If a structure has only two marker points, axial rotation (rotation around the line that connects the markers) cannot be measured. If this movement component is the object of study, a third marker point (obviously not in line with the other two markers) is necessary. I will only discuss the calculations for structures with two markers. The calculations with three markers are essentially the same. (With a third marker, two vectors can be defined for each structure, G1 and G2. Axial rotation is measured by calculating, in each frame, the component of G2 perpendicular to G1. The angle between these perpendicular components in subsequent frames is the axial rotation.)

To avoid an entirely abstract treatment of the method, it is illustrated by the movement of a gill arch of a white bream (Blicca bjoerkna). Two platinum markers were inserted in each gill arch, the copula communis (the fused basibranchials that connect the gill arches mid-ventrally) and the skull. The skull was the reference structure. All the above general requirements were fulfilled (for details and error analysis, see van den Berg et al. 1994).

The film-plane is the x,y-plane. The z-axis is perpendicular to this plane. All calculations in this paper are performed in this x,y,z-frame. The two markers in each structure define a vector, G. For example, G may represent a gill arch. One marker is translated to the origin (0,0,0). The coordinates of the other marker are (x,y,z). G can now be expressed in terms of x, y and z. Coordinates x and y are determined directly from each film frame (Fig. 1A,B). The value of z is calculated using Pythagoras’ rule (Fig. 1C):

Fig. 1.
(A) The markers in the fish head are indicated as small circles in this schematic film frame. The two black markers (at opposite ends of a gill arch) define a vector G (not in the film plane). (B) The x and y coordinates of G are calculated from its projection onto the film plane (=x,y-plane). (C) In this view, the film plane from B is shown from the side. The z coordinate of G is calculated with Pythagoras’ rule, given the length of G and the orientation of G (‘up’ or ‘down’) with respect to the film plane. The orientation of G (i.e. the sign of z) must be known. The wrong direction of G is indicated as G*.
View largeDownload slide

(A) The markers in the fish head are indicated as small circles in this schematic film frame. The two black markers (at opposite ends of a gill arch) define a vector G (not in the film plane). (B) The x and y coordinates of G are calculated from its projection onto the film plane (=x,y-plane). (C) In this view, the film plane from B is shown from the side. The z coordinate of G is calculated with Pythagoras’ rule, given the length of G and the orientation of G (‘up’ or ‘down’) with respect to the film plane. The orientation of G (i.e. the sign of z) must be known. The wrong direction of G is indicated as G*.

Fig. 1.
(A) The markers in the fish head are indicated as small circles in this schematic film frame. The two black markers (at opposite ends of a gill arch) define a vector G (not in the film plane). (B) The x and y coordinates of G are calculated from its projection onto the film plane (=x,y-plane). (C) In this view, the film plane from B is shown from the side. The z coordinate of G is calculated with Pythagoras’ rule, given the length of G and the orientation of G (‘up’ or ‘down’) with respect to the film plane. The orientation of G (i.e. the sign of z) must be known. The wrong direction of G is indicated as G*.
View largeDownload slide

(A) The markers in the fish head are indicated as small circles in this schematic film frame. The two black markers (at opposite ends of a gill arch) define a vector G (not in the film plane). (B) The x and y coordinates of G are calculated from its projection onto the film plane (=x,y-plane). (C) In this view, the film plane from B is shown from the side. The z coordinate of G is calculated with Pythagoras’ rule, given the length of G and the orientation of G (‘up’ or ‘down’) with respect to the film plane. The orientation of G (i.e. the sign of z) must be known. The wrong direction of G is indicated as G*.

formula
where G2 (=the length of G squared) and the sign of z are known (general requirements 3 and 4) (see Ellington, 1984).
When two structures are connected with a single joint (e.g. a gill arch and the copula communis), the three-dimensional angle α between these structures can be calculated as the angle between the vectors G1 and G2 representing these structures. If there is no marker exactly in the joint, the coordinates of the joint should be calculated using the coordinates of other marker points. The cosine of the angle a between G1 and G2 is given
formula
where x1,y1,z1 and x2,y2,z2 are the coordinates of vectors G1 and G2, respectively, and G1and G2 are the lengths of vectors G1and G2 (scalar).

The movement of a gill arch in a series of film frames (a film sequence) is the sum of its movement with respect to the skull and the movement of the skull with respect to the film frame. The separate components are interesting; their sum is not. Therefore, we want to separate these two components.

The movement of the skull can be split into a translation and a rotation component. The distance between the skull and the gill arches is not constant and is unknown. Therefore, the position (translation component) of the gill arches cannot be calculated relative to the skull in single-view films. However, rotation (e.g. depression) of the gill arches can be corrected for rotation of the skull.

The vector representing the skull in frame number n is Sn. The vector representing a gill arch is G. The angle between G and Sn can easily be calculated from equation 2. However, we want to know the depression angle of G, which is the angle between G and a horizontal plane (plane H) in the fish (Fig. 2A,B). The calculation of such a depression angle is more complicated. First, a frame in which the fish is horizontal is chosen as the reference frame (Fig. 2A). In this frame, plane H is parallel to the x,y-plane (or film plane) (by definition). All vectors in the other film frames (Fig. 2B) must be transformed to the orientation of the reference frame (the method is described below). The depression angle of G equals the angle between the corrected vector G and the x,y-plane, since the x,y-plane is always parallel to plane H after correction. The correction method is based on the movement of the skull vector Sn with respect to its reference orientation Sr. The direction of Sr should preferably be perpendicular to the film frame (see Appendix).

Fig. 2.
The depression angle of the gill arch vector G is calculated in a fish-bound frame by correcting G for rotation of the skull vector S. (A) Vector S in the reference frame is Sr. In the reference frame, plane H (the horizontal plane in the fish-bound frame) is parallel to the x,y-plane (by definition). (B) In film frame number n, the skull (vector Sn) has rotated with respect to Sr over an angle σ Plane H has also rotated over angle σ. The orientation of plane H with respect to vector S is unaltered. Vector G has to be transformed to the reference orientation given in A. (C) Vectors Sr and Sn define a plane P. This plane can have any orientation, depending on the way the skull has rotated (a combination of pitch, roll and yaw). G is projected on plane P (GP; step 1), rotated over angle σ (GPC; step 2) and restored to its original length (GC; step 3), by adding G-GP. GPC* is the wrong solution of GPC (rotated over angle −σ instead of σ).
View largeDownload slide

The depression angle of the gill arch vector G is calculated in a fish-bound frame by correcting G for rotation of the skull vector S. (A) Vector S in the reference frame is Sr. In the reference frame, plane H (the horizontal plane in the fish-bound frame) is parallel to the x,y-plane (by definition). (B) In film frame number n, the skull (vector Sn) has rotated with respect to Sr over an angle σ Plane H has also rotated over angle σ. The orientation of plane H with respect to vector S is unaltered. Vector G has to be transformed to the reference orientation given in A. (C) Vectors Sr and Sn define a plane P. This plane can have any orientation, depending on the way the skull has rotated (a combination of pitch, roll and yaw). G is projected on plane P (GP; step 1), rotated over angle σ (GPC; step 2) and restored to its original length (GC; step 3), by adding G-GP. GPC* is the wrong solution of GPC (rotated over angle −σ instead of σ).

Fig. 2.
The depression angle of the gill arch vector G is calculated in a fish-bound frame by correcting G for rotation of the skull vector S. (A) Vector S in the reference frame is Sr. In the reference frame, plane H (the horizontal plane in the fish-bound frame) is parallel to the x,y-plane (by definition). (B) In film frame number n, the skull (vector Sn) has rotated with respect to Sr over an angle σ Plane H has also rotated over angle σ. The orientation of plane H with respect to vector S is unaltered. Vector G has to be transformed to the reference orientation given in A. (C) Vectors Sr and Sn define a plane P. This plane can have any orientation, depending on the way the skull has rotated (a combination of pitch, roll and yaw). G is projected on plane P (GP; step 1), rotated over angle σ (GPC; step 2) and restored to its original length (GC; step 3), by adding G-GP. GPC* is the wrong solution of GPC (rotated over angle −σ instead of σ).
View largeDownload slide

The depression angle of the gill arch vector G is calculated in a fish-bound frame by correcting G for rotation of the skull vector S. (A) Vector S in the reference frame is Sr. In the reference frame, plane H (the horizontal plane in the fish-bound frame) is parallel to the x,y-plane (by definition). (B) In film frame number n, the skull (vector Sn) has rotated with respect to Sr over an angle σ Plane H has also rotated over angle σ. The orientation of plane H with respect to vector S is unaltered. Vector G has to be transformed to the reference orientation given in A. (C) Vectors Sr and Sn define a plane P. This plane can have any orientation, depending on the way the skull has rotated (a combination of pitch, roll and yaw). G is projected on plane P (GP; step 1), rotated over angle σ (GPC; step 2) and restored to its original length (GC; step 3), by adding G-GP. GPC* is the wrong solution of GPC (rotated over angle −σ instead of σ).

By positioning Sr and Sn tail-to-tail, a plane P can be defined (Fig. 2C). Plane P is the plane of movement of the skull; it is unrelated to the film plane. The amount of movement is expressed as the angle σ between Sr and Sn. Angle σ is a combination of the pitch, roll and yaw of the skull. Since Sr=Sn (requirement 3), cosσ is given by:
formula
In each film frame, G is transformed from the Sn orientation to the Sr orientation in three steps (Fig. 2C): step 1, G is projected on plane P (GP); step 2, GP is rotated over angle σ (GPC); step 3, using GPC, the corrected direction of G (GC) is calculated; note that GC=G.

When this is done, the depression angle of the gill arch is the angle between GC and the x,y-plane. Note that, in the calculations below, the coordinates are not transformed to a frame defined by plane P, but always remain defined in the original x,y,z-frame of the film plane.

Step 1: projection of vectorGon plane P. Just like any vector in plane P, vector GP must be a linear combination of Sr and Sn:
formula
where α1 and α2 are scalar factors.
GP is a perpendicular projection of G, therefore:
formula
Substituting equation 4a into equation 4b gives two equations with two unknowns (α1, α2):
formula
Using these equations, α1 and α2 and hence GP can be determined.

Step 2: rotation of vectorGPover angle σ. The coordinates of GPC (three unknowns: xPC, yPC, zPC) are calculated using three equations, which are based on three conditions for the rotation (see Fig. 2C): condition 1, GPC has the same length as GP; condition 2, GPC is rotated over angle σ; condition 3, GPC lies in plane P.

Condition 1: GPC=GP or:
formula
Condition 2: GPC is rotated over angle σ; combined with GPC=GP (condition 1):
formula
combined with equation 3:
formula
or:
formula
Condition 3: GPC lies in plane P; all vectors in plane P are perpendicular to SrXSn, therefore:
formula
or:
formula
Combination of equations 5, 6 and 7 yields a quadratic equation with two solutions for GPC. These solutions represent rotation over angle σ in both directions in plane P (Fig. 2C). The right solution is found by considering that the angle between Sr and GPC should equal the angle between Sn and GP (see Fig. 2C). Combined with Sr=Sn and GPC=GP we find:
formula
Step 3: restoringGPCto its original length
To put GPC ‘back in space’, we simply add the part of G that is perpendicular to plane P. This part, G-GP, is not affected by the rotation in plane P (Fig. 2C):
formula
Note that the effect on G of the above correction for skull rotation is dependent on the angle between G and plane P. When this angle is large, the effect of the correction is small. When the angle is 90°, its effect is even nil, since GC equals G.

MPW FORTRAN subroutines (for Macintosh computers) with the present calculations are available on request.

The example of the gill arch movements of white bream (van den Berg et al. 1994) illustrates the importance of the three-dimensional method of analysis. The abduction angle a between the left first gill arch and the copula communis was calculated using both the two-dimensional and the three-dimensional method. The projected angle αp (two-dimensional method) was 5–20° larger and had an amplitude two times larger (!) than the real angle a (three-dimensional method). The amplitude of depression angle uncor (the angle between the gill arch and the film plane) was about 1.5 times smaller than that of angle (the angle between the gill arch and a horizontal plane in the fish) because of the pitch of the fish during food intake. Furthermore, the data for βuncor suggested that the gill arch was placed in a special depressed position prior to gulping, while showed that this was an artefact.

The gill arch movement in our example consisted of a combination of abduction (angle α) and depression (angle β). The large differences between the two-dimensional and three-dimensional methods in the example clearly show that the latter method is essential for a quantitative analysis of such non-planar movements from single-view films. The three-dimensional method is also essential when substantial changes in the orientation of the animal occur. In the example, pitch was an integral part of the feeding behaviour of the white bream, which led to both quantitative and qualitative (e.g. so-called ‘special position of the gill arch’) errors when the two-dimensional method was used. The three-dimensional method allows us to calculate rotations in an object-bound frame. The three-dimensional method of analysis must be strongly advised for both quantitative and qualitative studies of animal movement.

     
  • G

    vector representing a gill arch

    •  
  • GP

    the projection of G on plane P

    •  
  • GPC

    GP corrected for rotation of vector S

    •  
  • GC

    G corrected for rotation of vector S

    •  
  • Plane H

    horizontal plane in the fish, which is parallel to the film plane when the fish is in the reference orientation

    •  
  • Plane P

    plane of movement of the skull, defined by vectors Sr and Sn

    •  
  • Film plane

    x,y-plane

    •  
  • S

    vector representing the skull

    •  
  • Sr

    reference orientation of S

    •  
  • Sn

    vector S in frame number n

    •  
  • x, y, z

    x, y and z coordinates of vector G

    •  
  • xP, yP, zP

    x, y and z coordinates of vector GP

    •  
  • xPC, yPC, zPC

    x, y and z coordinates of vector GPC

    •  
  • xC, yC, zC

    x, y and z coordinates of vector GC

    •  
  • xSr, ySr, zSr

    x, y and z coordinates of vector Sr

    •  
  • xSn, ySn, zSn

    x, y and z coordinates of vector Sn

    •  
  • xSrXSn, ySrXSn, zSrXSn

    x, y and z coordinates of vector SrXSn

    •  
  • α1, α2

    scalar factors to express GP in terms of Sr and Sn

    •  
  • α

    angle between gill arch and copula communis

    •  
  • αp

    the projection of angle a on the film plane

    •  
  • β

    angle between gill arch and plane H

    •  
  • βuncor

    angle between gill arch and the film plane

    •  
  • σ

    angle between vectors Sr and Sn

    •  
  • G·G

    notation for the dot product

    •  
  • G × G

    notation for the cross product

    •  
  • G

    notation for the length of a vector (=|G|)

Appendix

Vector Sr should be perpendicular to the film plane. There are two reasons for this. (1) The length of the projection of vector S on the film plane is the length S multiplied by the cosine of the angle between vector S and the film plane. The cosine is most sensitive to rotation when the angle is approximately 90°. This holds true for vector G as well: one should preferably film in the direction parallel to G, rather than perpendicular to it. In the latter case, the projection on the film plane is very insensitive to rotation of G (see Ellington, 1984, pp. 46–47). In other words, movements perpendicular to the film plane may easily go unnoticed in that case. (2) Axial rotation around vector Sr is not measured. When Sr is perpendicular to the film plane, this unmeasured rotation component is rotation in the film plane (yaw, in the example). During analysis of the film frames, both the marker projections and the outline of the fish head are copied on paper. The yaw component of head rotation can easily be compensated for by always positioning the outline of the fish head in the same way on the data tablet. Furthermore, if there is still some rotation in the film plane, this has no influence on depression angles.

Two anonymous referees are acknowledged for their thorough evaluation of the manuscript. This research was supported by the Foundation for Fundamental Biological Research (BION), which is subsidized by the Dutch Organization for Scientific Research (NWO), project number 811-428-265. Berend van den Berg is thanked for identifying the possibility of axial rotation around vector S. Mees Muller, Nand Sibbing, Jos van den Boogaart and Jan Osse are thanked for their critical comments on the manuscript.

Claes
,
G.
 and
De Vree
,
F.
 (
1991
).
Kinematics of the pharyngeal jaws during feeding in Oreochromis niloticus (Pisces, Perciformes)
.
J. Morph.
208
,
227
–
245
.
Google Scholar
Crossref
Search ADS
 
Drost
,
M. R.
 and
van den Boogaart
,
J. G. M.
 (
1986
).
A simple method for measuring the changing volume of small biological objects, illustrated by studies of suction feeding by fish larvae and of shrinkage due to histological fixation
.
J. Zool., Lond. A
209
,
239
–
249
.
Google Scholar
Crossref
Search ADS
 
Ellington
,
C. P.
 (
1984
).
The aerodynamics of hovering insect flight. III. Kinematics
.
Phil. Trans. R. Soc. Lond. B
305
,
41
–
78
.
Google Scholar
 
Hoogenboezem
,
W.
,
Sibbing
,
F. A.
,
Osse
,
J. W. M.
,
van den Boogaart
,
J. G. M.
,
Lammens
,
E. H. R. R.
 and
Terlouw
,
A.
 (
1990
).
X-ray measurements of gill-arch movements in filter-feeding bream, Abramis brama (Cyprinidae)
.
J. Fish Biol.
36
,
47
–
58
.
Google Scholar
Crossref
Search ADS
 
Nachtigall
,
W.
 (
1983
).
Biophysics of locomotion in air
. In
Biophysics
(ed.
W.
Hoppe
,
W.
Lohmann
,
H.
Markl
 and
H.
Ziegler
), pp.
601
–
609
.
Berlin, Heidelberg, New York, Tokyo
:
Springer-Verlag
.
Google Scholar
 
Sibbing
,
F. A.
 (
1982
).
Pharyngeal mastication and food transport in the carp (Cyprinus carpio L.): a cineradiographic and electromyographic study
.
J. Morph.
172
,
223
–
258
.
Google Scholar
Crossref
Search ADS
 
van den Berg
,
C.
,
van den Boogaart
,
J. G. M.
,
Sibbing
,
F. A.
 and
Osse
,
J. W. M.
 (
1994
).
Implications of gill arch movements for filter feeding: an X-ray cinematographical study of filter-feeding white bream (Blicca bjoerkna) and common bream (Abramis brama)
.
J. exp. Biol
.
191
,
257
–
282
.
Google Scholar
Crossref
Search ADS
PubMed
 
van Leeuwen
,
J. L.
 (
1984
).
A quantitative study of flow in prey capture of the rainbow trout, with general consideration of the actinopterygian feeding mechanism
.
Trans. zool. Soc., Lond.
37
,
171
–
227
.
Google Scholar
Crossref
Search ADS
 
Westneat
,
M. W.
 (
1990
).
Feeding mechanics of teleost fishes (Labridae; Perciformes): a test of four-bar linkage models
.
J. Morph.
205
,
269
–
296
.
Google Scholar
Crossref
Search ADS
 
Zarnack
,
W.
 (
1972
).
Flugbiophysik der Wanderheuschrecke (Locusta migratoria L.). I. Die Bewegungen der Vorderflügel
.
J. comp. Physiol.
78
,
356
–
395
.
Google Scholar
Crossref
Search ADS
 
©The Company of Biologists Limited
1994