This report will investigate how free-hanging objects act under the influence of gravity, in order to observe an aspect of a physics engine for a computer animation studio. To prove the concept, the shape of a hanging rope of length 205cm, supported at two ends that are 100cm apart was applied and mathematical functions were applied to it to determine the best model to duplicate the formation.
The functions were analysed and employed to form possible relationships that correspond to the relationship of the plotted points on the rope. These models will be generated through various methods to determine the best representation, through both technological and mathematic procedures, including simultaneous equation solving and technology, such as Excel and Desmos. The reliability and validity of the models will be tested through considering the correlation between the model and the photograph of the rope set and plotted points. Only even polynomials will be considered in this analysis, including quadratic, quartic, sum of even-degree polynomials and a piecewise function.
Variables:
The data collected to conduct the investigation will not be highly accurate or consistent, due to imprecise measurements since it was measured in centimetres which allows for error, and the lines forming the grid may be inaccurate and provide a source of major error. The rope type also affects the study as the density and thickness of rope (depending on type e.g. wire rope, synthetic) will vary results and the shape of the curve the rope falls in, therefore the function determined to model this specific curve may not suit other scenarios. When hung the rope was not completely consistent in how it fell, resulting in indents, misshaping the curve due to it retaining its previous rolled up shape. Therefore, these variables can be taken into consideration when observing the curve of the rope.
Assumptions:
It will be assumed that measurements of the rope were rounded to the closest centimetre. The curve of the rope will not be perfect and have faults due to variables, therefore will most likely be asymmetrical, with no exact perfect function to provide a best fit for the rope curve. The plotted points on the ropes curve will be rounded to one decimal point and was plotted as close to the centre of the rope. It was assumed that gravity is the only force acting on the rope and consequently created the turning point and a polynomial function.
Method:
A 100cm x100cm graph grid was drawn on a whiteboard, increasing in increments of 10cm. The experiment set-up is shown in Figure 1.
The bottom left-hand corner of the graph was chosen as the origin of the Cartesian plane. The 205cm long rope was affixed to the top corners of the cartesian plane at precisely the coordinates (0, 99.5) and (100, 98.7). The domain for the model with therefore be 0 x 98.7.
The image was opened in Desmos and scaled to the appropriate size of 100 units, the points were then recorded to one decimal place to match the middle of the rope to ensure accurate data was collected. The points were measured every 1 unit along the x-axis to ensure an accurate representation of the data was collected, providing 100 data points. Since the rope hangs freely due to its mass and the effect of gravity, forming an approximate symmetrical curve with a turning or apex point.
The 100 points were established into Cartesian coordinates relative to the origin. The data points are compiled in Table 1 Appendix 1. Plotting these junctures on the rope allowed these points to be implemented into residual formulas for the Desmos site to determine the best fitting expression is according to the points.
Data:
Figure 2 shows a plot of the data using excel, figure 3 shows the plot of data situated in Desmos in relation to the rope and figure 4 shows the plotted points without the photo reference. It was observed that the sketches resembled a parabola in shape.
Investigating Models:
Quadratic Functions
From figures 2, 3 and 4 it can be determined that the curve of the rope forms a parabola therefore, a quadratic function was investigated as it was assumed it would provide a rational function.
Using coordinates:
Left fixation coordinate (0, 99.5)
Right fixation coordinate (100, 98.7)
Turning point (100/2,20.7) = (50, 20.7)
Vertex Form Quadratic Function: y=a(x-h)2+k…(1)
The turning point for the curve of the rope is (50,20.7), showing that the curve is in the positive y- direction by 20.7cm and is 50 cm in the positive x-direction. Therefore, for values of h and k in equation (1), -50 and 20.7 can substituted in, resulting in:
The graph of model (A) and the plotted points is shown in Figure 5. It was also determined through Desmos technology that the model (A) had an R2 value of 0.9774, showing that it had a high resemblance to the rope curve however not perfect.
It was observed that although the curve travels through the two fixation coordinates and the turning point (as this was how the parabolic function was created), the curve made by the free hanging rope is flatter and wider than the parabolic function, is not best fitted to the plotted points, this could be improved by having a smaller ‘a’ value (affecting the dilation of the parabola).
Using Excel, a parabolic trendline was determine as depicted in Figure 6, this function found y=0.0322x2-3.3446x+104.65…Model (B). Although the R2 value is 0.9896 which indicates a strong positive correlation since it is close to 1, it is clear from the graph that the fit also does not correlate to the rope curve.
The analysis of the Quadratic formula was repeated using Demos tilde technology and the R2 value remain the same at 0.9896, however provided the equation:
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y=0.032245(x- 50.8636)2+17.9132 or y=0.032245x2-3.2802+101.33 (as seen in Figure 7) …Models (C).
Both Desmos equations depict the same function as the Excel produced equation, further confirming that this equation and function is considered by graphing technology as a substantial representation of the plotted points of the rope curve. However, it can be observed that it is not the ideal function to replicate the influence of gravity on a free hanging rope, as it has a lower turning point than the rope curve and does not pass through many plotted points. When compared the Desmos technology created a function with a higher R2 value than the mathematically determined function, as tilde allows for all plotted points to be considered rather than one point on the line that the mathematically determined function used. The Desmos Tilde technology function produced the best fitting quadratic function.
Quartic Functions
From Figures 5, 6 and 7 it can be observed that the curve appears flatter than the quadratic models, therefore quartic functions were considered and explored.
The graph of model (D) and the plotted points is shown in Figure 8. Through Desmos technology it was determined that the model (D) had an R2 value of 0.8311, showing that it had a low resemblance to the rope curve. It was observed that the although the curve travels through the two fixation coordinates and the turning point (as this was how the function was created), the curve has no similarities to the free hanging rope curve as it does not fit the plotted points.
It was therefore decided that to solidify that a Quartic function was not the best model to replicate the curve of a free hanging rope a tilde equation generated by Desmos was examined. It provided the equation:
y=0.000012608(x-50.6268)2+27.2695 (as seen in Figure 9) …Model (E).
The R2 value was 0.9555 which was higher than the mathematical equation, however it was 0.0341 lower than the R2 value for the quadratic tilde equations, thus a quartic function can be dismissed as a possible replication of the curve of a free hanging rope.
Refining Models:
Sum of Even-degree Polynomials
Through the exploration of Desmos it was concluded that even-degree polynomials always form a U-shaped curve (see Appendix 2), allowing a prediction that the curve of a free hanging rope would be fabricated through even-powered polynomials.
Thus, the equation: y =a0+b2(x-a2)2 +b4(x-a4)4 +…+b2n(x-a2n)2n +…
was utilized in exploring a function to replicate the plotted points of the rope curve.
It was than speculated that if a quartic and quadratic function was combined it may be the most suitable replica of the free hanging rope curve.
The graph of model (F) and the plotted points is shown in Figure 10. Desmos technology determined the R2 value for model (F) was 0.3161, showing that it had an extremely low resemblance to the rope curve.
It was observed that the function determined through simultaneous solutions is not the best fitted equation to replicate the free hanging rope curve as it does not share many similarities and only crosses 3 plotted points (2 of which were used in the simultaneous solution).
It was therefore decided that to solidify that a sum of even-degree polynomials function was not the best model replica a tilde equation generated by Desmos was examined. It provided the equation:
y=29.2529+0.0230967(x-50.5104)2+0.00000416876(x-51.2578)4 (as seen in Figure 11) …Model (G).
The R2 value was 0.9969, notably greater than model (F) R2 value, and is significantly closer to 1 than the quadratic function (Models B & C) R2 value (0.9896), showing that the technology generated equation is the better function to replicate the curve of a free hanging rope.
This technology generated equation provides a valid replication of a free hanging rope curve, which contradicts the equation calculated through simultaneous solutions. This is due to the tilde technology calibrating all plotted points to determine the best fit, whereas the simultaneous solution calculated was produced from three random plotted points and the vertex without taking into consideration the other points.
The Desmos generated equation could therefore be considered an appropriate model for the curve of a free hanging rope under the influence of gravity.
Piecewise Function:
A Piecewise function was investigated as it allows multiple sub-functions, each sub-function applying to a certain interval of the main function's domain. Hence, multiple functions could be created to suit the curve of the free hanging rope as precisely as possible in 6 functions.
A piecewise function was generated through trial and error on Desmos until 6 functions were created that followed and proceeds through majority of the plotted points. These collective functions generate a valid replica of a free hanging rope curve.
Conclusion:
The observation from the investigation in determining a rational function to model the curve of a free hanging rope influenced by gravity revealed that even degree polynomials were best suited as they always formed a U-shaped curve that imitated the rope curve. Many functions were explored to model the shape, the quadratic function had a R2 value relatively close to 1, the quartic function was than considered, however this proved invalid as it had a lower R2 value, therefore this was discarded. In appendix 4 and 5 a graph and table reinforce that all the models (including the sum of even-degree polynomials, however excluding the piecewise function) examined and analysed do not provide a perfect correlation between the plotted points on the rope and the function, as none of the residual illustrated display a y value of 0 and therefore disprove that they perfectly replicate the curve of the rope. In refining the model, the sum of even-degree polynomials and a piecewise function were examined for models of the free hanging rope. Through these different equations it was determined that the Desmos tilde equation had the highest established R2 value at 0.9969, however this is still not perfect, consequently it can be suggested that further analysis of multiple sum of even-degree polynomials could be examined to determine if another combination may result in a R2 value of 1 or higher than what was developed, as seen in Appendix 3 brief investigation of this was conducted. A piecewise function also provided a significant model of the rope, as it allowed small sections of the rope to be modelled by an equation allowing the function to pass through more plotted points, this provides a solution to matching the kinks and faults of the rope if more functions were utilized to gather a perfect and exact replica of the rope. Both the refining models; Desmos tilde sum of even-degree polynomials and piecewise functions provide adequate replicas of the curve of a free hanging rope influenced by gravity, however neither perfectly modelled the curve as points were excluded, therefore not precisely recreating the curve. The piecewise function however provides the opportunity to have a perfectly matching replica to the rope curve if more functions were determined, whereas as seen in Appendix 3 (Figure 16) it can be determined that there may not be a sum of even-degree polynomials that will gather a perfect R2 value of 1, thus it can be substantiated that a piecewise function would provide the most accurate replica of the curve of a free hanging rope under the influence of gravity.
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