This thread is about quantitatively measuring the jump height of different skaters. This post is about how to analyze videos of performances to do so, for anyone that might want to calculate a skater's jump height on their own. As you can imagine, it may be somewhat conceptually dense, though I think it's simply a matter of thinking through the various issues, and I try to lay out the reasoning in detail.
The TL;DR version is this: Take a video of the skater's jump, usually 25 FPS or 30 FPS. If it's 30 FPS, watch on frame-by-frame and look for identical (repeated) frames. If it has any, it's likely really just 25 FPS (with frames inserted). If it doesn't, then it's 30 FPS. Take the difference between the first frame where the skater is in the air, and the first frame where the skater lands (touches the ice). Then plug that number of frames into the following:
Height (in centimeters) = 981 * (frames/FPS/2)^2 / 2
For example, if the skater's first frame in the air was frame 2526, and the skater's first frame on the ice was frame 2539, then the number of frames was 13. For a 25 FPS video, this becomes:
Height = 981 * (13/25/2)^2 / 2 = 981 * (0.26)^2 / 2 = 33 cm
The error in measurement varies from 5 cm for a 25 FPS video to 2 cm for a 60 FPS video.
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A skater's jump height is an important indicator of how many rotations they can achieve in the air. The greater the height, the more time they have to rotate and thus the more rotations they can do. Height is also one bullet point for a jump's GOE (Grade Of Execution) score.
Making some basic assumptions about the skater, such as that air resistance is negligible and that the skater's legs are fully extended (or similarly extended) for takeoff and landing, the height can be calculated from the jump duration by the following:
height = (gravity) * ((duration/2)^2) / 2
This is based on the equation of motion for a body under uniform acceleration. The reason why the duration is divided in half is because the skater is moving upward for half of the jump's time, then vertically is stopped (regardless of the horizontal motion), then accelerates downwards for the latter half of the jump. The vertical distance traveled during the second half (which is the same as the distance during the first half) gives the height of the jump. Gravity is taken to be 9.81 m/s^2. So for example, if a skater is in the air for 0.6 seconds, then the height of the jump would be:
height = (9.81 m/s^2) * ((0.6 s/2)^2) / 2
= 9.81 * (0.3)^2 / 2
= 0.44 m or 44 cm
So although this is about measuring jump height, this is somewhat misleading. What is actually measured is the jump duration, and the jump height is then inferred from that.
The only remaining question, then, is how to measure how long the skater was in the air during the jump.
One way to measure the jump duration is by using a video of the skater's jump. Videos consist of image frames displayed one after another, usually with a set duration between each frame. A video's Frames Per Second (FPS) is the reciprocal of the duration. For example, a video that plays at 25 FPS has a duration of 0.04 seconds between each frame. The error of the measurement is related to the duration between each frame. Thus, a higher FPS means a lower error in measuring jump height. Assuming an error of +- 1 frame, the measurement error of a jump lasting about half a second, corresponding to a height of about 31 cm, for different FPS is below:
FPS Error
60 2 cm
50 2.4 cm
30 4 cm
25 5 cm
As you can see, the error is significant relative to the height at the lower FPS. 25 and 30 FPS are common. 50 FPS is rare, and is usually really just 25 FPS interlaced that has been de-interlaced. I will occasionally see 60 FPS but it's pretty rare, and usually means a very big video file.
To count the number of frames, take the difference between the first frame when the skater first leaves the ice (i.e. skates no longer touching the ice) and the first frame when the skater first lands (i.e. first skate touches the ice). Keep in mind that the skater actually took off sometime in between the last frame when the skate was touching the ice and the first frame when no skate is touching the ice, and also landed sometime in between the last frame when no skate is touching the ice and the first frame where the skate is touching the ice.
As a practical matter, determining this can sometimes be difficult, and I think this is where a lot of the arguments, err, internet discussion, can occur. For example, depending on the video resolution and brightness/contrast settings, it can be difficult to tell if the toe pick was still touching the ice or not in a given frame. Because of this, I tend to let one "inform" the other. For example, let's say a skater's toe pick was barely on the ice at frame 459 and clearly in the air at frame 460, and when I look at the landing, the skater was clearly in the air at frame 473 and clearly on the ice at frame 475, but I'm not sure if it was on the ice or not at frame 474. In this case, I'm more likely to use frame 475, because the skater (based on the takeoff) was in the air for longer than what the frame count would suggest. Thus, I try to match the conditions of the takeoff and jump: I try to match a "barely took off" with a "barely landed", and a "clearly took off" with a "clearly landed", when necessary.
Some other considerations for the video are its resolution and its brightness/contrast. Higher resolution means more detail can be seen, which makes it easier to determine when the skate has left the ice. Brightness and contrast affect whether or not you can judge this; a video that's too bright, or has too high of a contrast, will make the ice completely white, making it difficult to gauge if the skate is in contact with the ice or not. Obviously, the video should show the skater's skates, and the more zoomed in on the skater, the easier it is to judge if the skate is in contact with the ice.
Occasionally, the replays that are shown while waiting for the skater's scores can be used. But I find that in practice this is tricky; they can only be used if they are from the same camera, and are not simply the same footage with just repeated frames inserted. The math for using replays is similar (although it needs to take into account how much slower the replay is being run), except significantly more checks need to be done to make sure that the replay can be appropriately used. But occasionally they can provide very accurate determinations of a skater's jump height.
The TL;DR version is this: Take a video of the skater's jump, usually 25 FPS or 30 FPS. If it's 30 FPS, watch on frame-by-frame and look for identical (repeated) frames. If it has any, it's likely really just 25 FPS (with frames inserted). If it doesn't, then it's 30 FPS. Take the difference between the first frame where the skater is in the air, and the first frame where the skater lands (touches the ice). Then plug that number of frames into the following:
Height (in centimeters) = 981 * (frames/FPS/2)^2 / 2
For example, if the skater's first frame in the air was frame 2526, and the skater's first frame on the ice was frame 2539, then the number of frames was 13. For a 25 FPS video, this becomes:
Height = 981 * (13/25/2)^2 / 2 = 981 * (0.26)^2 / 2 = 33 cm
The error in measurement varies from 5 cm for a 25 FPS video to 2 cm for a 60 FPS video.
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A skater's jump height is an important indicator of how many rotations they can achieve in the air. The greater the height, the more time they have to rotate and thus the more rotations they can do. Height is also one bullet point for a jump's GOE (Grade Of Execution) score.
Making some basic assumptions about the skater, such as that air resistance is negligible and that the skater's legs are fully extended (or similarly extended) for takeoff and landing, the height can be calculated from the jump duration by the following:
height = (gravity) * ((duration/2)^2) / 2
This is based on the equation of motion for a body under uniform acceleration. The reason why the duration is divided in half is because the skater is moving upward for half of the jump's time, then vertically is stopped (regardless of the horizontal motion), then accelerates downwards for the latter half of the jump. The vertical distance traveled during the second half (which is the same as the distance during the first half) gives the height of the jump. Gravity is taken to be 9.81 m/s^2. So for example, if a skater is in the air for 0.6 seconds, then the height of the jump would be:
height = (9.81 m/s^2) * ((0.6 s/2)^2) / 2
= 9.81 * (0.3)^2 / 2
= 0.44 m or 44 cm
So although this is about measuring jump height, this is somewhat misleading. What is actually measured is the jump duration, and the jump height is then inferred from that.
The only remaining question, then, is how to measure how long the skater was in the air during the jump.
One way to measure the jump duration is by using a video of the skater's jump. Videos consist of image frames displayed one after another, usually with a set duration between each frame. A video's Frames Per Second (FPS) is the reciprocal of the duration. For example, a video that plays at 25 FPS has a duration of 0.04 seconds between each frame. The error of the measurement is related to the duration between each frame. Thus, a higher FPS means a lower error in measuring jump height. Assuming an error of +- 1 frame, the measurement error of a jump lasting about half a second, corresponding to a height of about 31 cm, for different FPS is below:
FPS Error
60 2 cm
50 2.4 cm
30 4 cm
25 5 cm
As you can see, the error is significant relative to the height at the lower FPS. 25 and 30 FPS are common. 50 FPS is rare, and is usually really just 25 FPS interlaced that has been de-interlaced. I will occasionally see 60 FPS but it's pretty rare, and usually means a very big video file.
To count the number of frames, take the difference between the first frame when the skater first leaves the ice (i.e. skates no longer touching the ice) and the first frame when the skater first lands (i.e. first skate touches the ice). Keep in mind that the skater actually took off sometime in between the last frame when the skate was touching the ice and the first frame when no skate is touching the ice, and also landed sometime in between the last frame when no skate is touching the ice and the first frame where the skate is touching the ice.
As a practical matter, determining this can sometimes be difficult, and I think this is where a lot of the arguments, err, internet discussion, can occur. For example, depending on the video resolution and brightness/contrast settings, it can be difficult to tell if the toe pick was still touching the ice or not in a given frame. Because of this, I tend to let one "inform" the other. For example, let's say a skater's toe pick was barely on the ice at frame 459 and clearly in the air at frame 460, and when I look at the landing, the skater was clearly in the air at frame 473 and clearly on the ice at frame 475, but I'm not sure if it was on the ice or not at frame 474. In this case, I'm more likely to use frame 475, because the skater (based on the takeoff) was in the air for longer than what the frame count would suggest. Thus, I try to match the conditions of the takeoff and jump: I try to match a "barely took off" with a "barely landed", and a "clearly took off" with a "clearly landed", when necessary.
Some other considerations for the video are its resolution and its brightness/contrast. Higher resolution means more detail can be seen, which makes it easier to determine when the skate has left the ice. Brightness and contrast affect whether or not you can judge this; a video that's too bright, or has too high of a contrast, will make the ice completely white, making it difficult to gauge if the skate is in contact with the ice or not. Obviously, the video should show the skater's skates, and the more zoomed in on the skater, the easier it is to judge if the skate is in contact with the ice.
Occasionally, the replays that are shown while waiting for the skater's scores can be used. But I find that in practice this is tricky; they can only be used if they are from the same camera, and are not simply the same footage with just repeated frames inserted. The math for using replays is similar (although it needs to take into account how much slower the replay is being run), except significantly more checks need to be done to make sure that the replay can be appropriately used. But occasionally they can provide very accurate determinations of a skater's jump height.