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This paper demonstrates that the pitch angle of a spiral galaxy can be calculated from the spiral arm simulation based on the new ROTASE model for the formation of spiral arms of galaxies, the new spiral equations from the model are more universal than other spiral formulas. A spiral arm length weighed average method is proposed to more fairly calc...
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Citations
... The following are the primary differential galactic spiral equations derived from the ROTASE model, please refer to the references [3,4] for the detail of derivation. ...
... For general application of the equations (1) for other than the simulation of spiral galaxies, the ρ can be treated as a normal parameter which is function of θ, and the R b is set to 1. A mini computer program can be written to calculate the x and y, and the calculated x and y must be rotated counter clockwise by the following equations (2) for nal spiral plotting, the counter clockwise rotation is well explained in the reference [4]: ...
Interesting graphic patterns can be created by mathematically manipulating the spirals calculated by the galactic spiral equations. Such mathematical manipulation is achieved by multiplying the spiral data with a special mathematical function named as art-pattern function. This method can be extended to mathematical equations other than spiral equations and to 3-dimension graphics. The graphic creativity is unlimited through this method. This method will have wide applications in many aspects of our daily life.
... Many mathematical spiral equations have been invented throughout the history of the science, like the Archimedean spiral, Euler spiral, Fermat's spiral, hyperbolic spiral, logarithmic spiral, Fibonacci spiral, etc. The author developed a set of new spiral equations in recent years based on the proposed Rotating Two Arm Sprinkler Emission model (for short, ROTASE model) [1][2][3][4], this model describes the possible mechanism of the formation of spiral arms of disc galaxies, regular galaxies with open spiral arms, 4 types of ring galaxies and galaxies with only one arm can be precisely simulated by the new galactic spiral equations, and several galaxies with special patterns are explained. However, when the author read a very general science article about interesting prime numbers, a strange idea or "whim" jumped to author's head when looked at the prime numbers in sequence (2,3,5,7….): ...
... The galactic spiral pattern will be decided by the behavior of the ρ. The calculated x and y must be rotated counterclockwise by the following equation for final spiral plotting, the counterclockwise rotation is well explained in the reference [4]: ...
In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P 1 = 1 + 6 * n (n > 0, n is an integer) P 5 = 5 + 6 * m (m ≥ 0, m is an integer) Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Twin prime numbers can only be formed between P1 and P5 prime numbers, n must be 1 greater than m. The largest prime number is known at the moment 2^(82,589,933)-1 is a P 1 prime number.
... Simulation of the phyllotactic spring-ring patterns seems a big challenge with those spiral equations mentioned above because none of those spiral equations can produce spiral-ring patterns. However, the author developed a set of galactic spiral equations in recent years from the proposed ROTASE model for the formation of spiral arms of barred galaxies [10,11]. The Preprints (www.preprints.org) ...
... The galactic spiral pattern will be decided by the behavior of the ρ. A mini computer program can be wriiten to calculate the x and y, and the calculated x and y must be rotated counterclockwise by the following equation for final spiral plotting, the counterclockwise rotation is well explained in the reference [11]: ...
This short paper demonstrates that the galactic spiral equations developed from ROTASE model can be used to morphologically simulate the phyllotactic spiral-ring patterns of plants. It is also noticed during this project that although the numbers in the Fibonacci sequence dominate the phyllotactic numbers, there are still significant phyllotactic numbers are not in the Fibonacci sequence, however, most of those numbers are well fit into a Fibonacci sibling sequence starting with numbers 0 and 2. Many plants show the phyllotactic number pairs of 10-16, 16-26 and 26-42 from the Fibonacci sibling sequence, which is the second dominated sequence in the phyllotaxis.
... The complementary filtering algorithm [14,15] can eliminate the accumulated error of the attitude sensor. When measuring the pitch angle of the road surface, a low-frequency accelerometer is added to detect the pitch angle of the vehicle when the vehicle is running [16,17], and the pitch angle calculated by the accelerometer is used to correct the measurement value of the attitude sensor. The high-frequency advantages of the attitude sensor and the low-frequency advantages of the accelerometer are effectively combined to reduce the measurement error when measuring alone. ...
The detection of long-distance pavement elevation undulation is the main data basis for pavement slope detection and flatness detection, and is also the data source for 3D modeling and quality evaluation of pavement surfaces. The traditional detection method is to use a level and manual coordination to measure; however, the detection accuracy is low and the detection speed is slow. In this paper, the high-speed non-contact vehicle-mounted road undulationelevation detection method is adopted, combined with the advantages of each sensor measurement; three methods are proposed to detect the road undulation elevation: rotary encoders, accelerometers, attitude sensor data fusion detection; GPS RTK detection; and Kalman filtering detection. Through modeling and experimental comparison, Kalman filter detection is not disturbed by the environment, and the detection accuracy is higher than the current international standard.
... Many mathematical spiral equations have been invented throughout the history of the science, like the A Archimedean spiral, Euler spiral, Fermat's spiral, hyperbolic spiral, logarithmic spiral, Fibonacci spiral, etc. The author developed a set of new spiral equations in recent years based on the proposed Rotating Two Arm Sprinkler Emission model (for short, ROTASE model) [1][2][3][4], this model describes the possible mechanism of the formation of spiral arms of disc galaxies, regular galaxies with open spiral arms, 4 types of ring galaxies and galaxies with only one arm can be precisely simulated by the new galactic spiral equations, and several galaxies with special patterns are explained. However, when the author read a very general science article about interesting prime numbers, a strange idea or "whim" jumped to author's head when looked at the prime numbers in sequence (2,3,5,7….): ...
... The spirals can be calculated by the differential equations (1) or the solution galactic spiral equations (2) to (5) by selecting the right equation according to the value of ρ. The calculated x and y must be rotated counterclockwise by the following equation for final spiral plotting, the counterclockwise rotation is well explained in the reference [4]: ...
In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. For the spirals only produced by the prime numbers in the 6-spiral pattern plotting, the spiral 2 and spiral 3 each has only one spiral point produced by prime number 2 and 3, respectively, all other spiral points produced by other prime numbers are located on the spiral 1 and spiral 5. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P1 = 1 + 6 * n (n > 0) P5 = 5 + 6 * n (n ≥ 0) In other words, every prime number greater than 3 is either a P1 prime number or a P5 prime number, no exception. Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Hope such sufficient condition can be found in the future. The number of P1 prime numbers roughly equal the number of P5 prime number in the first 2 billion prime numbers. The galactic spiral equations with golden angle can duplicate Vogel’s result for the simulation of sunflower seed head pattern, and a pinwheel pattern can be produced also with galactic spiral equations and 1 degree more than golden angle.
... Many special characteristics were also explained by this model. The average pitch angle of the spiral galaxies can be easily calculated after the successful spiral pattern simulation [16,17,18]. This paper will extend the application of the ROTASE model and address questions and confusions arose since the first publication of this model. ...
... The Rotating Two Arm Sprinkler Emission model can be briefly summarized below, and readers may refer the reference papers for the detail description of the model [16,17,18]. 1. ...
... The differential equation set (1) can be solved in the polar coordinate system in three different cases with four equations: ρ > 1, ρ = 1 and ρ < 1, respectively. Please refer the references for the detailed derivation [16,17,18]. For ρ < 1, the moving distance r of the X-matter to the galactic center is limited which results in the spiral ring pattern, the radius of the ring is defined by the following equation: ...
... The Rotating Two Arm Sprinkler Emission model can be briefly summarized below, and readers may refer the reference papers for the detail description of the model [16][17][18]. ...
... Many special characteristics were also explained by this model. The average pitch angle of the spiral galaxies can be easily calculated after the successful spiral pattern simulation [16][17][18]. This paper will extend the application of the ROTASE model and address questions and confusions arose since the first publication of this model. ...
... The differential equation set (1) can be solved in the polar coordinate system in three different cases with four equations: ρ > 1, ρ = 1 and ρ < 1, respectively. Please refer the references for the detailed derivation [16][17][18]. For ρ < 1, the moving distance r of the X-matter to the galactic center is limited which results in the spiral ring pattern, the radius of the ring is defined by the following equation: ...
This paper extends the application of the ROTASE model for the formation of spiral arms of disc galaxies, questions and confusions from readers about this model are addressed. The optical trail effect behind the spiral arm rotation is the natural consequence of the model. The morphologies of ring-galaxies are classified into four categories: type I: single ring; type II: 8-shaped double ring; type III: 8-shaped double ring wrapped by a larger outer ring; type IV: single ring without spiral and bar. All four types of ring galaxies can be described by the ROTASE model. The ROTASE model predicts that the false impression of spiral arm rotating ahead of the galactic bar in the galaxy MCG+00-04-051 will change with time, it will look like a normal galaxy with about 30° to 40° bar rotation in the future and the galactic bar ends will look like rotating ahead of the spiral arms with further 10 ° to 15 °bar rotation. The formation of one arm galaxies is due to X-matter at one side of supermassive black hole is much stronger than other side. More evidence is found to support the explanation of the formation and the evolution of the Hoag’s object. The possible evolution of spiral pattern of galaxies is illustrated by UGC 6093. The winding of the Milky Way could be tighter in the future based on the ROTASE model.
... Many special characteristics were also explained by this model. The average pitch angle of the spiral galaxies can be easily calculated after the successful spiral pattern simulation [16][17][18]. This paper will extend the application of the ROTASE model and address questions and confusions arose since the first publication of this model. ...
... The Rotating Two Arm Sprinkler Emission model can be briefly summarized below and readers may refer the reference paper for the detail description of the model [16][17][18]. ...
... The critical backward rotation is clearly explained in the reference paper [18]. Euler rotation may be carried out to match the orientation of the galactic disc to the line of the sight. ...
This paper extends the application of the ROTASE model for the formation of spiral arms of disc galaxies, questions and confusions from readers about this model are addressed. The optical trail effect behind the spiral arm rotation is the natural consequence of the model. The morphologies of ring-galaxies are classified into four categories: type I: single ring; type II: 8-shaped double ring; type III: 8-shaped double ring wrapped by a larger outer ring; type IV: single ring without spiral and bar. All four types of ring galaxies can be described by the ROTASE model. The ROTASE model predicts that the false impression of spiral arm rotating ahead of the galactic bar in the galaxy MCG+00-04-051 will change with time, it will look like a normal galaxy with about 30° to 40° bar rotation in the future and the galactic bar ends will look like rotating ahead of the spiral arms with further 10 ° to 15 °bar rotation. The formation of one arm galaxies is due to X-matter at one side of supermassive black hole is much stronger than other side. More evidence is found to support the explanation of the formation and the evolution of the Hoag’s object. The possible evolution of spiral pattern of galaxies is illustrated by UGC 6093. The winding of the Milky Way could be tighter in the future based on the ROTASE model.
... For example, if ρ > 1, select equation (5) to calculate the θ with variable r. r = R represents the current time. If ρ changes with time (rotation angle θ represents time), then use θ as variable to calculate ρ with equation (8), then, insert the θ and the calculated ρ into the right equation to find the matched r. 2. Use equation (4) to calculate the X and Y coordinates of the spiral arm: ...
... The backward rotation by equations (12) and (13) is critical as explained in detail in the reference [4] and will not be repeated here. Please note, for ρ < 1, when the r reaches its limit, continues to use equation (10) and (11) to calculate x and y with r(limit), and rotates x and y with equations (12) and (13), because θ keeps change, it will produce a beautiful spiral-ring pattern which has widely application in the nature, will be seen in the following sections Additional Euler rotation is needed for simulation of spiral galaxies to match the orientation of the line of sight. ...
... Its face profile shown in the image has a spiral-ring pattern similar to galaxy ESO 325-28, the spiral is extended with helix style like a thread, an unique and beautiful structure in the nature. Its face spiral pattern can be nicely simulated by the new spiral formulas (4) and (5) It seems that there is no good explanation why it takes this special shape, however, it must be the result of natural evolution through the history which makes it survived well. ...
A new set of spiral formulas is introduced as a new member of the spiral formulas’ family, people with interest in mathematics and natural spirals may use the new spiral formulas to simulate natural spiral objects or generate their own spirals. The new formulas are derived from a proposed new hypothesis called Rotating Two Arm Sprinkler Emission model (ROTASE) for the formation of spiral galaxies. In this paper, the derivation of the new spiral formulas is illustrated with boats moving across a circular river from central round island. The formulas have only one parameter and one variable, the parameter can change with time in any format, the morphology of the spiral is decided by the behavior of the parameter. 4 real spiral galaxies are precisely simulated shown as examples. It is demonstrated in this paper that the new spiral formulas can be also applied to simulate Earth natural objects such as plants, animals and hurricanes which have spiral patterns. The result shows that the new formulas seem more universal spiral formulas which can produce various different spirals patterns, can be applied to architecture, artworks and industry design. The new spiral formulas will be a new addition to the family of current mathematic spiral formulas.
Interesting graphic patterns can be created by mathematically manipulating the spirals calculated by the galactic spiral equations. Such mathematical manipulation is achieved by multiplying the spiral data with a special mathematical function named as art-pattern function. This method can be extended to parametric curve equations other than spiral equations and to 3-dimension graphics. The graphic creativity is unlimited through this method. This method will have wide applications in many aspects of our daily life.
