A Different Kind of Golf
By Daniel Kramer & Phil List
Mrs. Amy Bowman
Practical Applications of Advanced Mathematics
13 July 2001
The purpose of this project was to design a Disc golf course on the University of North Carolina Charlotte campus. Using the Internet to study the many aspects of the game, a Frisbee golf? course was designed. To design the course, we used a theodolite, a measuring tape, and trigonometric functions. The distance of each hole (from tee to hole) was calculated using the theodolite to measure angles and the measuring tape to find certain distances. Then, using trigonometric functions, the unknown distances were calculated. This yardage, along with any obstacles on the hole, was used to determine par for each hole.
Through extensive research, the history, mathematical aspects, and rules were discovered about this unique game, which has increased in popularity in recent years. In fact, there are over 16,000 PDGA members. The physics of the equipment are discussed, as well as the development. The findings and design of the course are discussed throughout the following paper.
Disc golf has been around for many years. It has developed into its modern form in the past 25
??years with the creation of the modern Frisbee and the Pole Hole. Prior to these breakthroughs, crude
discs and other improvised objects were thrown at random targets, or to people. With new technologies, the sport has grown. “Discs” are now used in place of Frisbees, and Pole Holes are being perfected
mathematically. Now there are tournaments with more than $50,000 in prize money.
Disc golf is played similarly to traditional golf. However, instead of a ball and clubs, players use a
???flying disc, or Frisbee. Disc Golf, Frisbee Golf, Dolf, & Frolf share similar principles. The sport was
formed in the 1970's, and shares with conventional golf the object of completing each hole in the fewest number of throws. A golf disc is thrown from a starting area, a tee, to a target, called the hole. The hole can
?be any number of disc golf targets; the most common is called a Pole Hole, an elevated metal basket. As a
player progresses down the fairway, that player must make each consecutive shot from the spot where the preceding throw came to a rest. The trees, shrubs, and terrain changes located in and around the fairways provide obstacles for the golfer. Finally, the hole is completed when the "putt" lands in the basket. Disc golf shares the same joys and frustrations of traditional golf, whether it involves sinking a long putt or hitting a tree halfway down the fairway. Another feature disc golf shares with traditional golf is that both are played in a beautiful setting (A guide).
There are some differences between disc golf and traditional golf. Disc golf rarely requires a greens fee, a cart won't be needed to rent, and no one will get stuck with a bad tee time. It is intended to be enjoyed by people of all ages, male and female, regardless of economic status. Disc golf can be played from school age to old age, making it the one of the best lifetime fitness sports around. Specially-abled and
disabled people participate, giving them the opportunity to take part in a mainstream activity. Since disc golf is easy to learn, no one is excluded. Players match their pace to their capabilities, and proceed. The Professional Disc Golf Association is the governing body for the sport, and sanctions competitive events for men and women of every skill level, from novice to professional. Nowadays, there are over 16,000 members of the PDGA, and permanent disc golf courses are found in countries worldwide, as well as throughout the U.S. (A guide).
Many city parks have golf courses already arranged. Most are free to play. Disc golfers who do not have the advantage of a permanent disc golf facility in their area often create their own “courses” in nearby parks and green spaces. A nine-hole disc golf course can be established on as little as five acres of land, and a championship-caliber 18-hole course on 30-40 acres. Disc golf courses can coexist with existing park facilities and activity areas. The perfect location combines wooded and open areas, and a variety of terrain changes (A guide).
Disc golf has evolved since the dawn of mankind. The early cavemen, in their search for weapons to extend their ability to slay prey, probably used rocks before clubs. Killing something from a distance was considerably safer than with a club or a sharp stick. Flat rocks have a different flight path and fly further than other weapons; for example, skipping flat stones on the water or throwing shields. Then came the discus that Discoblus threw which resembled a Frisbee? in its shape. In the early steel age, sharpened rings where thrown to cause injury, which they accomplished with devastating effect. They flew with accuracy, caused serious injury and looked like the modern Aerobie? (it is akin to a Frisbee? with a hole in it). Then came the ancient word "scaling" (to throw a thin flat object), so that its edge cuts through the air. Pie pans, film can lids and toy flying saucers were the recent predecessors of the modern Frisbee? (An abbreviated history).
In 1964, Ed Headrick was employed by Wham-O. In his first three months on the job, he developed and filed a mechanical patent on his version of what a flying saucer ought to look like. During the same period, he thought of a marketing plan and made the first test mold out of an old disc mold that Wham-O had acquired. The greatness of the invention lain in its simplicity, and it was quickly learned that there was a hard core of people in their 20's--perhaps 100 people in the world, a “Saucer Cult”--that were playing with
a child's toy, a flying saucer, and loving it. In 1964, Headrick offered them a "Pro" model, white with a black flame painted ring, a gold foil label that said “108 grams”, and the Olympic rings upside down. It looked strange, and the saucer cult loved it. Hence, his claim to have developed the modern Frisbee?. Then he formed the International Frisbee Association (IFA) the same year and started shooting his first television commercial called "What's a Frisbee?" The modern Frisbee patent was licensed to Wham-O as required by his contract (33 years).
Since that time, the Frisbee?, made under the teachings of the patent with the "Lines of Headrick,”
has sold over 200,000,000--a stack from California to New York and back 6.3 times, end to end. The IFA had over 112,000 members who all shared the following: the love, the companionship, and the camaraderie of a piece of plastic. Ed Headrick was CEO and sales exceeded 18 million with earnings in the millions (33 years).
Soon, many people were playing Frisbee golf. A game where people would throw an expensive Frisbee into the ground every throw on purpose? Wow! What a market potential! It seemed so easy, but what could possibly be better than walking through a beautiful park and throwing at trees, drinking fountains, open car windows, and an occasional coed? In the early 1970s, Headrick and Co. went back to the drawing boards and, 56 models later, a contraption was born to catch a Frisbee. Shazam! Chain! Chains were the “holy grail” of this revolutionary sport. Chain, it is indestructible, flexible, and makes a pleasant sound when struck. Chain was the answer (33 years). Headrick invented the first Disc Pole Hole, called the Mach I, consisting of 10 chains hanging in a parabolic shape over an upward opening basket (US Patent 4,039,189; issued 1975). The game was formalized when Ed Headrick designed and installed the first formal Disc Golf Course was that same year in Oak Grove Park, Pasadena, California. It was an instant success. He also founded the Professional Disc Golf Association in 1975. In 1983, Headrick was too old & gray to run the sport, so he turned it over to the players to run. Since that time, Disc Golf has been transformed because of our natural competitive nature (An abbreviated history). Following the Mach I came the Mach models II, III, V through 20+ years of blood, sweat and tears (33 years).
Millions of people are now having fun damaging discs playing the game of Disc Golf. The sport is on the eve of an exponential growth. Perhaps the world can find peace through the communication of a piece of plastic and chains (33 years). What would Gandhi have thought?
This research shows that much work has gone into the physics of a PDGA approved Disc Pole Hole?. The main objective of this target is to catch a Frisbee like a human hand. The following is a summary of what was discovered over the years as the Disc Pole Hole? has evolved.
The Disc Pole Hole? will have at least one steel ring to attach the chains to a small circle around the central pipe, thus preventing the chains from flaying out of control when struck by a Frisbee?. The rings served the purpose too well; as all of the replicas that use chain have copied this feature. Once a disc strikes one chain, the weight of 18 links is lifted, absorbing the energy of the impacting disc, plus the weight of the ring, and even the weight of several adjacent links attached to the ring on each side of the impacted chain. If you add these neighboring chains that the disc must also strike, the amount of resistance against the disc increases exponentially. This function of the chains being gathered by a ring works well when the
strikes the exact center of the chains that are being held by the ring or rings. The disc is almost always caught because the forces of the chain & the disc are exactly symmetrical; nothing is pushing the disc away from its flight path (New Chains).
The problems start when the disc impact is uneven, since the disc has struck the chains off to the side, not the center. Some of the same exponential force that once stopped the disc are now pushing the disc away from being caught by the basket. It does so with a force that is increasing exponentially as the disc impacts more hanging chains on its elliptical escape path. There are also complications caused by the disc spinning, i.e. a right-hand spin that strikes the right side will pull itself into the chains, where as an impact to the left side will actually ricochet itself away from center. The opposite occurs when the spin is left-handed (New Chains).
To recapitulate this phenomenon, it is believed by professionals that the minimum amount of
should be the maximum energy required to stop the forward motion of the energy required to catch a disc
disc and allow gravity to pull it into the basket. Therefore, it was concluded that the steel ring system has too much kinetic energy available, so it must be changed in order to meet the goal of the catching device to catch like a human hand (New Chains).
Testing of the characteristic of loose chain has indicated that the
amount of energy to sufficiently stop a disc is contained in a section of chain
consisting of roughly 18 links and having a mass of about 239g. To
guarantee its ability to stop a disc thrown at a high speed, approximately 3
links (with a mass of 39.256g) were hung from the inner chain and attached
in a loop where they can be lifted by the 18-link chain, as required. The outer
chains may be attached to sliding hooks, like the teachings of the Mach V
patent, or to the ridged hooks of the Mach III patent. The 12 inner chains
form a cylinder like a Mach III and return to the top chain holder parallel to the pipe. The 12 outer chains are joined at the juncture of the upward hanging 3 links by an S-hook (or other suitable device) to the downward hanging chains forming a loose, regular dodecagon of 12 sides of approximately 8" in diameter. This allowed a maximum weight to stop a disc of 21 links or 278 g under all conditions, multiplied by the
number of chains involved but in no case more than 60 links or 804 g. All chains are fettered in this manner and one chain is still of sufficient mass to catch a disc on the right side, right hand throw, or 3 chains on the left side (results may differ depending on the coefficient of friction of the disc and chains). Compared to the maximum weight of 390 g on a Mach III available to absorb the energy of a disc, the 278g on this invention represents a major break through in the efficiency of this catching device and makes a soft and gentle stop of the disc. This change allows the disc to fall into the basket, rather than ricocheting into the basket with enough energy to cause it to bounce out, thereby missing an otherwise perfect putt (New Chains).
A real advantage of this chain arrangement is that the outer chains, because of their lighter mass, hang almost straight down, as opposed to the parabolic shape of the heavier system. The filling in of this area, when applied to 3 different Pole Hole models, is beneficial. It adds more 22% to the catching area of a MACH III and a 15% increase on the catching area of both the MACH II and MACH V (New Chains). The preceding was the culmination of the physics & technology of Frisbee golf?.
Now that some of the math involved in Frisbee golf? is understood, it is time to move on to the other aspects of the game. The need for more courses is continuous, as the popularity of Frisbee golf? continues to grow. The PDGA formed standards for the design and installation of new golf courses, to ensure their success with the public. The ongoing fitness boom encourages more and more people to take up recreational activities to improve their health and quality of life. Disc golf? provides upper and lower body conditioning, aerobic exercise, and promotes a combination of physical and mental abilities that permit very little risk of physical injury. Concentration skills also improve by mastering shots and negotiating obstacles. Players of limited fitness levels can start slowly, and then progressively increase their level of play as their fitness improves. Scheduling is also flexible; a round takes one to two hours, and may be played alone, eliminating the difficulty of scheduling tee times. Disc golf provides yearlong fitness, even in rain or snow. Perhaps the greatest characteristic of the sport is the expense - or rather, the lack thereof. A professional quality disc costs less than $10, and it only takes one for basic play. Like with traditional golf, disc golfers find themselves addicted, increasing the prospect of frequent participation (PDGA, 2001).
Headrick, Ed. 33 years with the Frisbee?: A retrospective of the past and future of disc golf. Disc
Golf Association. Retrieved 9 July 2001 on the World Wide Web at www.discgolfassoc.com under The History of Disc Golf
-----. The new chains. Disc Golf Association. Retrieved 9 July 2001 on the World Wide Web at www.discgolfassoc.com
An abbreviated history of disc golf 1 million B.C. to present day. Disc Golf Association. Retrieved 9
July 2001 on the World Wide Web at www.discgolfassoc.com
A guide to disc golf from the PDGA. Professional Disc Golf Association. Retrieved 9 July 2001 on the
World Wide Web at www.PDGA.com