Journal Logo

Applied Sciences: Biodynamics

3-D biomechanical analysis of sprint hurdles at different competitive levels


Author Information
Medicine & Science in Sports & Exercise: February 1997 - Volume 29 - Issue 2 - p 231-237
  • Free


In recent years top level athletic competition has become more difficult owing to the increasing demands of the events. Athletes must improve all their capacities to the optimum for the chance to give a winning performance. One key factor for development is technique, particularly in those events that demand a high level of skill. Thus, the biomechanical analysis of different events and different skill levels within events helps in the understanding of critical points of technique.

Examination of the literature revealed several two-dimensional kinematic and kinetic studies for the hurdles event, especially sprint hurdles(3,6,8-11). However, most of these studies show data from training situations. Because of the nature of the athletes, the demand for competitive challenge is needed to perform at an optimum level. Thus, it is important to collect data from competitive situations (4,7) despite the problem associated with competition studies of the limited number of subjects and trials available for analysis.

Mann and Herman (4) suggested that the horizontal velocity of the center of mass (CM) during the clearance is the best indicator of hurdle performance. It is evident from the literature(4,6,7,9,11) that the value of horizontal velocity varies from 6.16 m·s-1 to 9.27 m·s-1, which indicate different genders, level of competition, and circumstances of study. The other important indicator is vertical velocity at take-off. Mann and Herman (4) presented values of 1.35, 1.68, and 1.76 m·s-1 for the female gold, silver, and 8th place performers in the 1984 Olympic Games, respectively, and Schlüter(11) measured a range of 2.0-2.4 m·s-1 for the nine international and national level decathletes in the training situation.

All these studies have been concerned primarily with two-dimensional analysis. However, the development of motion analysis systems has made it easier to carry out three-dimensional (3-D) studies, which can provide more accurate information on both rotational and translational variables common in the hurdle clearance situation. McDonald and Dapena (5) have succeeded in combining a 3-D analysis in high level competition with a relatively large number of subjects, showing that average horizontal velocities of the CM during hurdle clearance were 8.33 ± 0.11 m·s-1 and 8.57 ± 0.23 m·s-1 for the nine females and 23 males, respectively.

A long take-off distance to the hurdle has been recommended as it allows the athlete to maintain a high horizontal velocity (6). Review of the literature shows that a variation in stride length and distances at the different parts of clearance exists (Table 1). However, different analysis systems and definitions of variables are used, and hence the comparisons among studies should be carried out with care.

Since the take-off distance is farther from the hurdle than the landing distance, one would assume that the peak of CM parabola occurs before the hurdle. However, since the CM is in front of the take-off foot and almost directly above the landing foot, at take-off and landing, respectively, the peak of the parabola is closer to the hurdle than expected by considering distances measured from foot positions. Mero and Luhtanen(6) measured the peak of the flight path parabola at 0.46 m before the hurdle with one tall elite athlete. In the World Championships in 1983, they (7) measured horizontal distances of 0.00 and 0.16 m from the peak of the CM path to the hurdle for two males and reported a range of 0.22-0.59 m for three females. McDonald and Dapena(5) found that the peak of the CM path occurred 0.03± 0.15 m and 0.30 ± 0.16 m before the hurdle for the elite male and female athletes, respectively. They (5) concluded that the results show that male athletes do not have to move the peak of the CM path nearer to the take-off point to have enough space for the movement of the lead leg, which is contrary to suggestions by others.

In most studies the number of subjects used for analyses has been limited or the subject groups have been homogeneous. Thus, the purpose of this study was to investigate the hurdle clearance at different competitive levels of performance in male and female athletes during competition situations.


The videotaping carried out for this study was performed at British National Championships and County Competitions between December 1992 and January 1994.

Data contains videotaping of one indoor (60-m hurdles) and one outdoor(100-m hurdles) session for females and three indoor (60-m hurdles) sessions for males. Written consent was obtained for the analysis and videotaping process in this study. Two video camera recorders (JVC GY-X1 using S-VHS video tape) operating at the rate of 25 Hz, thus yielding 50 fields per second were positioned on the same side of the track with the best possible view considering the stands and other obstacles in the field.

The cameras were beside the track, one to the front of the hurdle and the other toward the rear side of the hurdle. The camera's optical axes were approximately 90° to each other from the view of action. All data were collected from the clearance of the third hurdle (both male and female). The third hurdle was chosen because the difference in position between the third hurdles of female and male indoor races is 2.00 m (30.00 and 32.00 m from the starting line for female and male, respectively), thus making it possible to videotape this hurdle from both events without moving the camera positions.

The cameras were genlocked to each other to ensure simultaneous exposure. The exposure times were 1/500 s or 1/1000 s depending on lighting conditions. The Peak Performance 24-point calibration frame (1.60 × 1.92 × 2.22 m, width, height, and length, respectively) was used for calibrating the analyzed volume of each lane. The calibration frame was located at the place of the third hurdle horizontally and vertically by spirit level and parallel with lane lines. After the calibration of position one, the frame was moved through the next lanes to optimize the number of athletes digitized from one race. Male and female hurdle positions were calibrated separately when both sexes were analyzed from the same competition. The “Kine Analysis” digitization package presented by Bartlett (2) controls for accurate calibration by testing the digitized calibration frame against a standard computed from the actual calibration frame values. Inaccurate calibration is not acceptable and must be redigitized.

The fields of the selected trials were chosen and digitized using an Arvis(Manchester, U.K.) digitization board interfaced to an Acorn Archimedes 440 microcomputer (Cambridge, U.K.) using a 14-segment model construction. Two additional points (both upper corners of hurdle crossbar) giving the reference of the hurdle height were included in the model. The Direct Linear Transformation (DLT) algorithm (1) was used for reconstructing the real world coordinates from digitized data of the two views. The smoothing and data calculations were carried out using the cross-validated quintic spline (12).

A total of 28 trials were analyzed yielding 14 clearances each for men and women. The trials were further subdivided into four groups according to the official times from the competition. Group M1 consisted of seven male county level performances (8.2-8.4 s in indoors 60-m hurdles). Group M2 had seven male club level indoor performances (8.6-9.2 s). Group F1 consisted of six outdoor female performances (13.08-13.72 s in 100-m hurdles) plus one indoor trial (8.4 s) to form the female elite level group. Group F2 had seven female county level performances indoors (8.9-9.6 s). The mean ± SD of age, height, and mass of athletes are presented in Table 2.

The data presented in this study are concerned with the kinematic variables of hurdle clearance from the midstance phase of take-off contact to the midstance phase of landing contact (the midstance phase is determined for the first field when CM horizontally passed the toe coordinate of the supporting leg). Mann and Herman (4) and Rash et al.(9) found that some hurdlers had their whole body CM in front of the foot CM at landing contact. The current study did not use the foot CM but instead used the toe coordinate to measure distances. In this study the whole body CM was not in front of the toe coordinate at the beginning of landing contact in any subjects.

The videotaped view was restricted to the hurdle clearance phase from take-off to landing, which incorporated all lanes except the outer lanes. This was done to optimize image size and number of subjects, thus increasing the accuracy of the digitization and analysis process. However, concentration for the middle lanes does limit the number of the potential subjects. The limitations of a 50-Hz analysis precluded inclusion of certain variables requiring more exact timing or higher sampling rate.

Selected values were collected from Kine Analysis software and typed into an Excel 2.1 software package (Macintosh IIsi microcomputer, Cupertino, CA) where variables, which needed further determination (i.e., deviation angle at take-off and mean horizontal velocity of CM), were calculated. Finally all variables (26 in total) were transferred to Minitab 8.2 Statistical Software package running on the Apple Macintosh computer. A Pearsons' Product Moment correlation coefficient within groups and an independent t-test(with Bonferonni inequality adjustment for alpha levels P = 0.00064) between male groups, female groups, and sexes were performed. An overall test(i.e., MANOVA or DM MANOVA) could not be performed because of the low number of subjects in each group.

Description of variables used in the results. Race time is the official result of the competition (hand timing on indoor with an accuracy of 0.1 s and electrical timing on outdoor with an accuracy of 0.01 s). Mean horizontal velocity of center of mass is the displacement the CM traveled horizontally from the midstance contact phase before the hurdle to the midstance contact phase after the hurdle divided by the time taken for this movement. The midstance contact phase is determined for the first field when CM passed the toe coordinate of the supporting leg. CM lift is the vertical displacement of CM from the midstance contact phase before the hurdle to peak of the CM path during flight.

Variables of CM velocities at take-off and landing are vertical velocity at take-off, vertical velocity at landing, horizontal velocity at take-off, and horizontal velocity at landing. These variables are determined from the first field after toe-off and the last field before ground contact at take-off and landing, respectively.

Other variables considered for the take-off and landing are the take-off angle, which is the angle against the horizontal plane made by the CM calculated from the coordinates of CM between the last field with toe contact and the first field after the toe-off; the deviation angle at take-off, which is the angle between the CM and the toe of the supporting leg against the horizontal plane (Fig. 1) and which is calculated from the coordinates of the CM and toe of the supporting leg as an average of two fields, the last field with toe contact and the first field after the toe-off; take-off distance, the horizontal displacement from the toe to the hurdle at the last field of contact; and landing distance, the horizontal displacement from the hurdle to the toe at the first field of contact.

The remaining variables are the horizontal distance of CM to the hurdle at peak of the CM path during flight; clearance height, the height of CM to the hurdle bar when the CM pass the hurdle; the minimum hip angle of lead leg during clearance; the maximal angular velocity of the trail hip during clearance; the timing of maximal angular velocity of trail hip, time from take-off (the first field after toe-off) to the field when the angular velocity of the trail hip reached maximum; total stride length, the summation of take-off and landing distances; and flight time, the time from the first field after the toe-off before the hurdle to the first field with the toe contact with the ground after the hurdle.


Level of performance. The average (± SD) mean horizontal velocities of CM during the clearance were 7.8 ± 0.1, 7.3 ± 0.2, 8.3 ± 0.4 and 7.0 ± 0.3 m·s-1 for the groups M1, M2, F1, and F2, respectively. In other studies hurdling velocities varied considerably. For example, Mann and Herman (4) measured horizontal velocities of the hurdle clearance from 7.73 to 8.33 m·s-1 for the three female finalists in 1984 Olympic Games, Mero and Luhtanen (6) showed 8.26 m·s-1 for a top male hurdler in training, and Schlüter (11) measured an average mean velocity of 7.4 m·s-1 for 9 decathletes in the clearance of first hurdle. From the World Championships (1983) Mero and Luhtanen (7) presented an average velocity of 8.21-8.84 m·s-1 for three females and 8.92-9.27 m·s-1 for two male hurdlers in the semifinals. Rash et al. (9) filmed six of the top 10 American female 100-m hurdlers who performed the range of horizontal velocity from 6.16 m·s-1 to 7.94 m·s-1 in a simulated competition situation. Also, McDonald and Dapena (5) measured an average 8.33 m·s-1 for nine females and 8.57 m·s-1 for 23 male competitors during the 1988 U.S. Olympic trials. Thus, it can be concluded that the female subjects in this study represent International and County levels while the male subjects represent County and Club levels.

The male group M2 reached 94% of the mean velocity for the male group M1(P = 0.014), whereas the female group F2 reached only 85% of the velocity for the better female group F1 (P = 0.0000). This produced more statistically significant differences between the female groups than between the male groups, reflecting the larger spread of ability levels in the female groups.

Comparison of male groups. A graphical representation of the means and SDs for the take-off distance, take-off angle, horizontal velocity at take-off, and horizontal distance of CM to the hurdle at the highest point of flight path are presented in Figure 2 for the male groups. There were no statistically significant differences between the male groups in these variables. Concerning the CM path, group M1 raised their CM 0.27 ± 0.04 m from the midstance phase to the highest point of clearance, whereas the same value for the group M2 was 0.33 ± 0.04 m. The vertical movement of CM is considered one of the more important variables in describing hurdling performance (3,4). Further, no statistically significant difference was evident in the vertical velocities at take-off between the male M1 and M2 groups (1.7 ± 0.2 m·s-1 and 1.9 ± 0.2 m·s-1, respectively,P = 0.110). In absolute terms these results show that group M1 performed with a reduced vertical CM lift.

Schlüter (11) measured values as high as 2.3± 0.1 m·s-1 for vertical velocity at take-off for nine decathletes. The results of this study are in better agreement with McDonald and Dapena (5), who reported 1.76 ± 0.13 m·s-1 in the same variable although their male athletes were at higher level than the subjects in this study. For a constant resultant velocity, a higher vertical velocity results in a loss of horizontal velocity. However, from McDonald and Dapena's data and the data from this study, it appears that in the men's sprint hurdles approximately 1.7 m·s-1 vertical velocity is needed to clear the hurdle successfully. Once the athlete has reached this level, the differences among athletes could be seen from examination of other variables.

CM lift and vertical velocity of the CM indicate indirectly the force production of the leg extensor muscles and the direction of propulsion. It is possible to reduce the CM lift by trying to keep the body in an upper position(straightened joint angles) during the last stride and by producing less vertical velocity at take-off. To make these improvements, the athlete needs to strengthen his/her leg and hip muscles and to gain more skill and confidence to horizontally approach the hurdle. An increase in muscle power resists flexion of the leg joints, which is why it is possible to have a flatter CM path during the contact phase and then after the take-off.

Comparison of female groups. A graphical representation of the means and SDs for the take-off distance, take-off angle, horizontal velocity at take-off, and horizontal distance of CM to the hurdle at the highest point of flight path is presented in Figure 3 for the female groups. These graphs illustrate the relationships between the variables at the beginning of the clearance phase. The longer take-off distance allows a lower take-off angle, which results in greater horizontal velocity. This allows the athlete to have the peak of the CM flight path substantially before the hurdle(Fig. 3.4).

In addition to the variables mentioned above, the female groups differed markedly in the timing of maximal angular velocity of the trail hip after the take-off and in the horizontal velocity at landing (P = 0.0001). The angular velocity of the trail hip reached a mean maximum 0.13 ± 0.03 s after the take-off in the group F1 and after 0.16 ± 0.02 s in the group F2. It is assumed that this indicates that the better athletes can perform quicker movements of the body (technique and skill).

Comparison of males and females. There were no significant differences between the men and women subjects when comparing mean horizontal velocity of the CM. Thus the differences between the sexes could be primarily a result of the difference in the hurdle height if one looks only at hurdle clearance. The descriptive statistics for the stride pattern of the clearance for the total male and female groups as well as the probability of differences between the sexes are presented in Table 3. These variables are in agreement with those in other studies(4,5,7).

The sexes differed from each other also in the vertical velocities at take-off. The males' vertical velocity at take-off was 1.8 ± 0.3 m·s-1, whereas the vertical velocity at take-off for the females was 1.6 ± 0.2 m·s-1. When a difference at take-off occurred, it was also natural to have a difference in the vertical velocity of landing. The landing vertical velocities were -1.4 ± 0.2 m·s-1 and -1.2 ± 0.2 m·s-1 for males and females, respectively. McDonald and Dapena (5) had slightly lower values for the females take-off vertical velocity (1.49± 0.14) and slightly higher landing velocity values (-1.53 ± 0.18 m·s-1 and -1.43 ± 0.13 m·s-1) for the male and female groups, respectively. However, considering the different methods of calculation and the use of different motion analysis systems, these results are in good agreement.

The minimum hip angle of the lead leg was 45 ± 7° for men and 61± 7° for women (P = 0.0000) in this study. The male hurdlers leaned more with their upper bodies and raised their lead legs more than the female hurdlers, permitting the male athletes to lower their CM parabola. Although negligible, this method also reduces air resistance. The mean maximal angular velocities of trail hip flexion were 748 ± 119°·s-1 and 638 ± 66°·s-1 for the men and women, respectively.

The female average CM lift was 0.23 ± 0.03 m, while the male average was 0.30 ± 0.05 m (P = 0.0003). This reduced value of CM lift for the female group still gave an elevated flight path for their CM, which could be seen in the clearance height (fig. 4.1). Interestingly, however, the women had a larger margin over the hurdle than the men. McDonald and Dapena (5) found similar results. The larger variations in the vertical path of the CM lead to a greater amount of wasted energy.

McDonald and Dapena (5) showed that it is not advisable for women to clear the hurdle with a low CM path because they would need longer intersteps between hurdles which could have a detrimental effect on the running speed. These calculations appear logical, however, if the athlete lowers the path of CM, the horizontal velocity should increase. When the horizontal velocity increases and when the length of the intersteps are not maximum in comparison with a sprint of the same speed, it should be possible to have the longer intersteps with a flatter parabolic flight path. This should enable females to clear the hurdle with a lower parabola than measured in this study and by other researchers.

Furthermore, McDonald and Dapena (5) suggested that a lower parabolic flight path with a shortened flight phase would disturb the legs in preparing for a favorable landing position. This can be questioned because movements associated with hurdling are not at the limits of human capacity. Hence, it should be possible for athletes to develop faster movements to accommodate the shorter flight phase.

On average, the female subjects in U.S. Olympic trials in 1988 reached the peak of the parabola 0.30 m before the hurdle, while the male subjects reached their peaks almost directly over the hurdle (5). As can be seen in Figure 4.2, the results in this study show the same trend although the SDs are larger.

The total horizontal displacement of the CM in the flight phase was split into two parts by the parabola peak. These relative portions of the CM path were examined and the average peak of parabola for the male groups occurred horizontally slightly farther in relation to take-off than for the female groups (54 ± 4% and 55 ± 6% for male M1 and M2 groups and 54± 7% and 51 ± 3% for the female groups F1 and F2, respectively). However, there were no statistically significant differences between either the male groups or the female groups or the sexes. This shows that the relative peaks are at a similar point in the parabolas. The reason that the female group F1 has an absolute peak before the hurdle is that their take-off distance is greater compared with the other female group(Figs. 3.1 and 3.4). This gives them the space to approach the hurdle horizontally.

It is important to note that with a velocity of 8.0 m·s-1 the CM is moving 0.16 m from field to field when sampling at 50 Hz which means 5-6% from the total clearance distance of CM. Thus within this error limit, it is possible that the peak of the parabola is in the middle of clearance, although it is unlikely because the reduced videotaping rate does not influence the path of the CM to the degree that it affects other variables such as certain joint angles.

Correlation analysis. The statistically significant correlations(Pearsons Product Moment) identified for mean horizontal velocity of CM for different groups are presented in Table 4. Interestingly, the only significant correlation between race time and mean horizontal velocity was found in the male group M2. However, the authors believe that the mean horizontal velocity of CM better represents the skill of hurdle clearance than does the race time because it is possible to make errors in other parts of the event, which could influence race time. Correlations in female groups F1 and F2, respectively, for the take-off distance and CM lift against mean horizontal velocity, provide information regarding the importance of the long take-off and the shape of parabola discussed previously.

If one considers the parabolic path of CM, an interesting question is why it appears that the male athletes do not need the space for their horizontal approach on the hurdle. In the male group M2, the correlation between the location of CM peak and mean horizontal velocity of the CM was negative; furthermore, some of these athletes had this CM peak after the hurdle. Biomechanically, this is not an ideal situation and further research into this area is needed.

The deviation angle at take-off (Female group F1 in Table 4.) shows that the less acute (the more towards 90°) the angle between the toe and the CM the higher the mean horizontal velocity. Body position at take-off could explain this finding. Thus, three more variables (trunk position, hip extension of the trail leg, and CM height) relating to deviation angle were examined to investigate the specific reasoning behind this finding. However, because there were no significant correlations of the trunk position or hip extension of the trail leg, these findings are not supported by the conclusion of body position. Nevertheless, the height of the CM had statistically significant correlation with the deviation angle, but the correlation was negative: the lower the CM position the greater the angle (r =-0.871, P < 0.05, N = 7). Thus it seems that taller athletes with longer legs could lean more forward in take-off, which for this variable is less important considering the total performance. It could be that the best athletes are the best in spite of their mechanics rather than because of their mechanics. However, it is recommended that studies on this variable involving more subjects and subject groups, together with subjects of different heights and variables in relation to CM height, be carried out before final conclusions are presented.


Based on their times, the better female athletes in this study showed an International level of performance, and the differentiation of performance levels between the female groups was larger than between the male groups. This produced a greater number of statistically significant differences between the female groups than between the male groups. Owing to the limitations of this study and of motion analysis systems it could be concluded that: 1) In the men's sprint hurdles approximately 1.7 m·s-1 vertical velocity is needed to clear the hurdle successfully. However, once the athlete has reached that level, the differences between athletes are a consideration of other variables. 2) The better female athletes approach the hurdle more acutely (towards a flat position), farther away, and with greater horizontal velocity than the lower level female group. 3) The reason for the difference in position of peak of parabolic flight path for the male and female groups is unclear and needs further investigation.

It is important to carry out biomechanical studies both in competition and training situations, particularly in the event of sprint hurdles. However, it is important to increase the number of subjects studied at different levels of performance to improve statistical validity. Furthermore, the demands placed on the athletes with different anthropometric compositions should be addressed in future research.

Figure 1-Deviation angle at take-off.
Figure 1-Deviation angle at take-off.
Figure 2-Take-off distance (2.1), take-off angle (2.2), horizontal velocity at take-off (2.3) and horizontal distance of CM to the hurdle at the peak of parabolic path (2.4) for male groups. Mean ± SD values and the probability of
Figure 2-Take-off distance (2.1), take-off angle (2.2), horizontal velocity at take-off (2.3) and horizontal distance of CM to the hurdle at the peak of parabolic path (2.4) for male groups. Mean ± SD values and the probability of :
t -test between groups are presented. N = 7 in both groups. Bonferonni inequality adjustment for alpha levels P = 0.00064.
Figure 3-Take-off distance (3.1), take-off angle (3.2), horizontal velocity at take-off (3.3) and horizontal distance of CM to the hurdle at the peak of parabolic path (3.4) for female groups. Mean ± SD values and the probability of
Figure 3-Take-off distance (3.1), take-off angle (3.2), horizontal velocity at take-off (3.3) and horizontal distance of CM to the hurdle at the peak of parabolic path (3.4) for female groups. Mean ± SD values and the probability of :
t -test between groups are presented. N = 7 in both groups. Bonferonni inequality adjustment for alpha levels P = 0.00064.
Figure 4-Clearance height (4.1) and horizontal distance of CM to the hurdle at the peak of parabolic path (4.2) for combined male and female groups. Mean ± SD values and the probability of
Figure 4-Clearance height (4.1) and horizontal distance of CM to the hurdle at the peak of parabolic path (4.2) for combined male and female groups. Mean ± SD values and the probability of:
t -test between groups are presented. N = 14 in both groups. Bonferonni inequality adjustment for alpha levels P = 0.00064.


1. Abdel-Aziz, Y. I. and M. Karara. Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry. In: Proceedings of the ASP/IU Symposium on Close-Range Photogrammetry, Urbana, IL Falls Church, VA: American Society of Photogrammetry, 1971, pp.1-18.
2. Bartlett, R. M. The definition, design, implementation and use of a comprehensive sports biomechanics software package for the Acorn Archimedes 440 micro-computer. In: Proceedings of the VIIIth International Symposium of the Society of Biomechanics in Sports, M. Nosek, M., D. Sojka, W. E. Morrison, and P. Susanka (Eds.). Prague: Conex Company, 1990, pp. 273-278.
3. Kollath, E. Zum Einfluss einzelner Körpersegmente auf die vertikale Abfluggeschwindigkeit des KSP beim 100-m-Hürdenlauf.Leistungssport 13(4):37-43, 1983.
4. Mann, R. and J. Herman. Kinematic analysis of Olympic hurdle performance: Women's 100 meters. Int. J. Sport Biomech. 1:163-173, 1985.
5. McDonald, G. and J. Dapena. Linear kinematics of the men's 110-m and women's 100-m hurdles races. Med. Sci. Sport Exerc. 23:1382-1391, 1991.
6. Mero, A. and P. Luhtanen. A biomechanical analysis of top hurdling. Mod. Ath. Coach 22(July):3-6, 1984.
7. Mero, A. and P. Luhtanen. Biomechanische Untersuchung des Hürdenlaufs während der Weltmeisterschaften in Helsinki.Leistungssport 16(1):42-43, 1986.
8. Mero, A. and E. Peltola. Elektromyographische Aktivität und Reaktionskräfte beim Hürdenlauf.Leistungssport 19(4):29-31, 1989.
9. Rash, G. S., J. Garrett, and M. Voisin. Kinematic analysis of top American female 100-meter hurdles. Int. J. Sport Biomech. 6:386-393, 1990.
10. Salo, A., E. Peltola, and J. T. Viitasalo. Einige biomechanische Merkmale des Zwischenhürdenlaufs im 110 m-Hürdenlauf.Leistungssport 23:59-62, 1993.
11. Schlüter, W. Kinematische Merkmale der 110-m-Hürdentechnik. Leistungssport 11(2):118-127, 1981.
12. Wahba, G. and S. Wold. A completely automated French curve: fitting spline functions by cross-validation. Commun. Statist. 4:1-17, 1975.


©1997The American College of Sports Medicine