Aim
The aim of the folio is to further demonstrate our ability to accurately apply the mathematical concepts and relationships that have been learnt in class, to explore a mathematical model for the features of Bezier curves.
Introduction
A Bezier curve is a parametric curve defined by a minimum of three points consisting of a starting and terminating point and a control point which alters the shape of the curve. The curve passes through the starting and terminating points and is defined for . In this folio Bezier curves with 4 points were used, consisting of a starting and terminating point and 2 control points (C_{1} and C_{2}), the curve is defined by the following parametric equations:
This folio will first look at the origins and real life application uses of Bezier curves. Then will require a real life object for instant a building to be accurately modeled by using a number of separate Bezier curves, giving the parametric equations for each curve. All to increase our knowledge of the properties of Bezier curves and how they function.
Method
Task one consists of researching the origins of Bezier curves, hence who were they created by and their initial use. In addition to this research on their current real life application uses was required, giving a number of examples of where Bezier curves are used and explaining the significance of their roles.
Task two of the folio required us to investigate an object which could be modeled by Bezier curves, for instant a building which was used in this folio. Initially the dimensions of the structure had to be determined by research and approximating using a ruler and ratios. Once the dimensions of the structure was established the starting and terminating points could be determined, thus using the computer graphing tool Desmos a basic Bezier curve with the desired starting and terminating points was constructed. From there the shape of the curve was change by moving the control points until the shape matched the structure being modeled, then by using the equations mentioned above the paramedic equations of the curve could be calculated. Descriptions to why the specific shape was chosen was made. Once all the Bezier curves equations were calculated they were graphed using Desmos once again and the shape was compared to the real life structure.
Research: Origins of Bezier curves and their application
Pierre Etienne Bezier was born on September the 1^{st}, 1910 in Paris. He was a mechanical and electrical engineer, who in 1933 at age 23 began working for car manufacture Renault initially as a tool setter. However, during 1960 Bezier had made substantial ground at Renault being elected manager of technical development which mainly focused on the body design of motor vehicles (Bezier curves, 2016). During 1960 Bezier also began to research CAD/Computer-aided Manufacturing, with the intentions of drawing machines and motor vehicle bodies parts using free formed curves (Education, 2015). In 1968 UNISURF was launched which used curves with parametric polynomials to help model automotive bodies. These curves soon becoming the know ‘Bezier curves’. However, there is some speculation to who really invented these types of curves. French physicist and mathematician Paul de Casteljau, calmed he developed the first curves of this type in 1959 using de Casteljau’s algorithmand a numerically stable to evaluate the curves. Bezier just apparently made the curves popular and gave a use for them in manufacturing according to Paul de Casteljau (Engineer, Pierre Bezier, inventor of the Bezier curves, 2006).
Currently, Bezier curves have a number of real life application uses. The first use was developed by Bezier himself, using the curves to help model motor vehicle bodies, due to the smoothness achieved by the specific curves and the ability for them to be easily manipulated. His UNISURF programs are still being used currently. Another real life application is type design, which uses complex Bezier curves of generally 3^{rd} to 5^{th}degrees, or a large number of small separate curves are used to create fonts for computer programs. As well as being vastly used in graphic design for logos (Gross, 2016). An example of Bezier curves being used for font design is shown in Figure 1.
Another real life application Bezier curves are used in, is engineering. Bezier curves are used to model bridges and buildings due to their ability to model perfectly smooth curves that conventional functions cannot produce (Gross, 2016). An example of a bridge which used Bezier curves to engineer its shape is shown in Figure 2.
Modeling Situation: Sydney Opera House
This section will investigate making a mathematical model of a structure using multiple Bezier curves, specifically the Sydney Opera House shown in Figure 3.
After researching the dimensions of the structure it was found that the highest point of the roof (left peak) was 65m. However, a number of approximations and assumptions were made in terms of the dimensions of the structure due to a lack of information. Using approximated ratios, the other dimensions such at the height of the second peak and the total length was calculated. Initially the length from peak to peak was determined so the starting and terminating points of the overall curve could be found. The total length from peak to peak was approximately 1 and a half times longer than the know height, thus 100m. The length of the base was then established via the same method, the base was the approximately the same length as the known height therefore, 65m. The second peaks height was then calculated, it was approximately two thirds the length of the known height, so 40m. Lastly the position of intersection between the two curves were determined, an assumption was made of 30m high and 60m across form the heights peak. Once these points were determined the following diagram was drawn to give representation of the rough position of all starting and termination points, shown in Figure 4. To create the structure 4 separate Bezier curves were used.
Bezier Curve: A
Curve A is the side wall for the highest peak of 65m, hence the starting point was made to be (0,65). From Figure 4, the point at which the wall hits the base was determined to be 15m, hence the terminating point is (15,0). Once the starting and terminating points were positioned the two control points were manipulated until the desired shape was achieved. The shape as seen in Figure 3 has a slight curve however is relatively straight, hence the control points were moved close to the curve to achieve the straight yet slightly curved line. From Figure 5, control point 1 was approximated to (13,43) and control point 2 was approximated to (15,19) for ease of calculations. From these 4 points the following parametric equations were calculated using the Bezier curve equations mentioned in section 1:
Bezier Curve:
Curve B is the Curve for the largest and highest section of the roof, having a peak of 65m, hence the starting point was made to be (0,65) to meet the side wall. From Figure 4, the point at which the roof stops (intersects other roof section) was determined to be 30m high and 60m along from the highest peak, hence the terminating point is (60,30). After the starting and terminating points were positioned the two control points were changed, changing the shape until the desired shape was achieved. The shape as seen in Figure 3 for curve B has the greatest curve of all the parts which is a smooth continuous curve, therefore to achieve the smooth curve the control points were evenly positioned apart. From Figure 6, control point 1 was approximated to (23,64) and control point 2 was approximated to (45,53) for ease of calculations. From these 4 points the following parametric equations were calculated using the Bezier curve equations mentioned in section 1:
Bezier Curve: C
Curve C starts form the point of intersection between both roofs, which from Figure 4 was determined to be 30m high and 60m along from the initial peak. Therefore, the starting point (60,30) was used. This specific curve is the smallest out of the two with a maximum height of 40m and being 100m away from the initial peak, thus the terminating point (100,40) was used. Once the starting and terminating points were positioned the two control points were changed, changing the slope and position of maximum until the desired shape was achieved. The shape as seen in Figure 3 shows that curve C is the same as curve B however it is slightly tilted having the maximum point in the middle of the curve instead of at the peak. Thus the curve must have a maximum before the terminating point. From Figure 6, control point 1 was kept exactly at (69.5,39) to ensure the correct curve, however control point 2 was approximated to be (82,43) for ease of calculations. From these 4 points the following parametric equations were calculated using the Bezier curve equations mentioned in section 1:
Bezier Curve: D
Curve D starts from the base of the structure (x axis), which from Figure 4 was determined to be 80m along from the y-axis. Therefore, the starting point (80,0) was used. Curve D is the side wall for the small curved roof (curve C) so the maximum height is 40m and is 100m away from the initial peak, thus the terminating point (100,40) was used. Once the starting and terminating points were positioned the two control points were moved until the desired shape was achieved. The shape as seen in Figure 3 shows that curve D is the straightest of all four curves with only a slight curve. Hence, similar to curve A the control points were positioned close to the resulting curve to create a straight line with a slight curve. From Figure 8, control point 1 was approximated to be (83,12) and control point 2 was approximated to be (91,30) for ease of calculations. From these 4 points the following parametric equations were calculated using the Bezier curve equations mentioned in section 1:
Complete Model
Once all four Bezier curves have been modeled and their parametric equations have been calculated, the whole model of the Sydney Opera house can be constructed by graphing the four Bezier curves on the same set of axis. As shown below:
As seen in Figure 8 the model constructed by the four Bezier curves is very close to the actual dimensions and shape of the real Sydney Opera House. However, it can be seen that the point at which the two roofs intersect is slightly lower and to the left of the model, thus the terminating point of curve B and starting point of curve C are incorrect in terms of the real life structure. In addition, the small roof is shorter than the real model.
Discussion
Due to the investigation using a computer generated graphing tool there were very few limitations throughout the procedure because any functions modeled were easily drawn with accuracy. However, one limitation to Desmos appeared when investigation the control points the values given were only to 1 decimal point, hence lowering the accuracy. Another limitation was the fact that only one correct measurement of the structure could be found, that being the height of the tallest peak. This limited the accuracy of the model due to all other measurements were approximations found by ratios, this also indicates the first assumption made. During the folio all measurements besides the height of the tallest peak were assumed to be correct using ratio methods and visual inspections, where in reality they were not accurate as seen in Figure 9, leading to a slight difference in size compared to the real structure. In addition to this we were assuming that approximating each control point to the nearest whole integer would not affect the overall shape of the curve, this lowered the accuracy of the model due to not using the true values of each control point. Lastly it is assumed that when we look at the Bezier curves produced on Desmos the correct values and shapes are being displayed. Does this need more detail?
Conclusion
Consequently, from investigating the shape and dimensions as well as making approximations of the structure, a realistic and accurate model of the Sydney Opera House was able to be constructed. This is further confirmed by Figure 9.
Appendix