Angles Test Answer Key
Deidamia Bassiti
I stumbled upon this seemingly difficult question while doing a Mensa practice test. I have since been trying to figure it out but I can't. I am calling it seemingly difficult because these questions always appear to be difficult, until you come up or are provided with, their simple and elegant answer.
I am converting angles-axis representation to Euler angles. I decided to check and make sure that the Euler angles I got from the conversion would lead back to the original axis-angle. I print out the values, but they do not match! I have read =5408 and _between_quaternions_and_Euler_angles as well as similar conversion questions on this website.
The rotations are in order Z*Y*X as is common, I believe. Is there someting wrong with my math? I plan on eventually adding special cases for when pitch is 0 or PI as in but right now I think the problem is separate.
If you want to test your code, I suggest checking orthogonal rotations of the unit basis vectors. For example rotate [1, 0, 0] with 90 degrees appropriately to get [0, 1, 0]. Check whether you really get the expected [0, 1, 0], etc. If you got the rotations of all 3 basis vectors right then your code is most likely correct.
If your subject of rotation cannot roll then using quaternions to represent your rotation is Incorrect. Quaternions encode 3 dimensions of rotation into the system and if your system has only two the representation is mismatched and incorrect.
If you only have pitch and yaw. Quaternion transformations can give you an answer involving roll which is fundamentally incorrect. Zeroing the roll angle does not prevent your transformations from having a roll value. The Quaternion is not encoded with the concept of rotations Without a roll so it is incorrect to use it here.
For things that can only pitch and yaw and NOT roll, use entities that do not involve the concept of "rolling" like 3D Cartesian coordinates or spherical coordinates (not euler angles). That is enough and more correct. You will not suffer from gimbal lock under these conditions... using quaternions for this is not just over kill but WRONG.
If you can't remember that formula, simply divide the shape into triangles. The sum of the interior angles in each triangle measures 180 degrees, so for each triangle add 180 degrees and you get the sum of all the angles in the polygon.
The polygon in this problem has 4 sides, so you know its interior angles add up to 360 degrees. The problem tells you that the sum of angles a and b is more than 70 degrees. The lowest possible value for a + b is 71 degrees. If
Lisa Zimmer Hatch, MA, and Scott A. Hatch, JD, have been helping students excel on standardized tests and navigate the college admissions process since 1987. They have written curricula and taught students internationally through live lectures, online forums, DVDs, and independent study.
Adolescent idiopathic scoliosis affects 1% to 3% of U.S. adolescents. It is defined by a lateral curvature of the spine (Cobb angle) of at least 10 degrees in the absence of underlying congenital or neuromuscular abnormalities. Adolescent idiopathic scoliosis may be detected via the forward bend test and should be confirmed with scoliometer measurement. Mild scoliosis is usually asymptomatic; it may contribute to musculoskeletal back pain, but there is no evidence that it causes disability or functional impairment. Patients with severe scoliosis (Cobb angle of 40 degrees or more) may have physical pain, cosmetic deformity, psychosocial distress, or, rarely, pulmonary disorders. Several studies have shown modest benefit from bracing and scoliosis-specific physical therapy to limit progression in mild to moderate scoliosis, but there were no effects on quality of life. Because no high-quality studies have proven that surgery is superior to bracing or observation, it should be reserved for severe cases. There is little evidence that treatments improve patient-oriented outcomes. The U.S. Preventive Services Task Force and the American Academy of Family Physicians found insufficient evidence to assess the balance of benefits and harms of screening for adolescent idiopathic scoliosis in children and adolescents 10 to 18 years of age.
Adolescent idiopathic scoliosis is a lateral curvature of the spine (i.e., the Cobb angle) of 10 degrees or more that affects adolescents 10 to 18 years of age. It is the most common form of scoliosis and is distinguished from other types of scoliosis by the absence of underlying congenital or neuromuscular abnormalities. Approximately 1% to 3% of adolescents in the United States are affected.1 The incidence is similar between males and females. However, females are 10 times more likely to progress to Cobb angles of 30 degrees or more.1,2 Genetic factors are thought to contribute to the development of scoliosis, but inheritance patterns are variable, and no single mode of genetic transmission has been identified.2 There is no role for genetic testing in the screening and management of adolescent idiopathic scoliosis.
Although most patients with adolescent idiopathic scoliosis are asymptomatic, a small minority develop clinical symptoms such as physical discomfort and respiratory compromise with social and psychological consequences such as cosmetic deformity and psychological distress, all of which can reduce quality of life.
Most patients with adolescent idiopathic scoliosis will not develop clinical symptoms in their lifetime. Back pain is more common among those with scoliosis; however, it does not result in disability or functional impairment. Curve magnitude does not correlate with back pain severity. Mortality is similar between patients with adolescent idiopathic scoliosis and the general population.3
A minority of patients who have greater curve angles will develop substantial rib deformities, leading to more serious disease later in life. Clinically significant disease is more likely to occur at a Cobb angle of 40 degrees or more.3,4 The incidence of spinal curvature greater than 40 degrees is 0.4% among adolescents 10 to 16 years of age.1 Those with progressive scoliosis may have physical pain, cosmetic deformity, and respiratory difficulties ranging from subjective shortness of breath to pulmonary disorders with measurably impaired lung function. These diseases are accompanied by other adverse effects, including social and psychological distress, financial cost, and reduced quality of life.2,3
The U.S. Preventive Services Task Force (USPSTF) and the American Academy of Family Physicians (AAFP) found insufficient evidence to assess the balance of benefits and harms of screening for adolescent idiopathic scoliosis in children and adolescents 10 to 18 years of age.2,3
The USPSTF found adequate evidence that adolescent idiopathic scoliosis can be accurately detected in asymptomatic adolescents using three screening techniques: the forward bend test, scoliometer measurement, and Moir topography. Although bracing may slow curve progression in mild to moderate scoliosis, there is insufficient evidence that early detection and treatment improve health outcomes. No published studies have investigated the direct harms of screening or interventions.3
Because curve progression occurs primarily with skeletal growth during adolescence, proponents of screening have argued that early detection and treatment could slow curve progression before skeletal maturity and improve long-term health outcomes.3,4 This was the rationale for nationwide school-based scoliosis screening programs for decades. However, in 2004 the USPSTF and AAFP recommended against routine screenings, citing a lack of evidence that benefits exceed potential harms, including exposure of low-risk adolescents to unnecessary radiography, referrals, and bracing.5 Other professional organizations, including the American Academy of Orthopedic Surgeons, the Scoliosis Research Society, the Pediatric Orthopedic Society of North America, and the American Academy of Pediatrics, support routine screenings, arguing that early detection improves the chances of effective nonsurgical management and citing the relatively low cost and minimal radiation risk associated with screening. They suggest screening females twice, at 10 and 12 years of age, and males once at 13 to 14 years of age.6
Scoliosis is often brought to the attention of clinicians by the patient, caregivers, or school-based screening programs. Clinical examination including the forward bend test and use of a scoliometer can reliably determine which patients require further evaluation or referral.
Patients with adolescent idiopathic scoliosis are typically identified through school-based screening programs or when the patient, caregivers, or clinicians notice a curve or asymmetry (e.g., asymmetric breasts, chest wall, shoulders, or back). There are no reliable physical symptoms of scoliosis. Significant back pain is uncommon except in severe disease and should cause the examiner to suspect alternative diagnoses such as infection, inflammation, or neoplasm.2 Restrictive or obstructive pulmonary disease can result from severe scoliosis but will not likely be the cause for presentation. Clinical examination can identify patients who require imaging for calculation of the Cobb angle or for follow-up.
Surgical referral is generally recommended for patients with a Cobb angle of at least 40 degrees. Those with lesser curvatures can be monitored unless there are signs or symptoms suggesting an alternative diagnosis.
No evidence-based guidelines exist to help determine which patients require orthopedic referral for adolescent idiopathic scoliosis. Referral should be considered for those with severe scoliosis or curvature and musculoskeletal symptoms suggestive of an alternative diagnosis. In general, few referrals are made because of a lack of evidence that treatment affects long-term outcomes, except in severe cases.
Referral for surgery and/or bracing is indicated when the initial Cobb angle is 40 degrees or more.2 Observation is generally recommended for patients with an initial Cobb angle less than 20 degrees, although referral for physical therapy may be appropriate.3 Patients with a smaller Cobb angle and risk factors for progression may also benefit from referral or close observation. Younger age at presentation (especially younger than 12 years), female sex, family history of clinically significant scoliosis, and relative skeletal immaturity are risk factors for progression.2,14,17 Skeletal maturity is generally estimated through radiography of the pelvis to obtain the Risser grade. Grading is based on the degree of iliac apophysis ossification, from grade 0 (no ossification) to grade 5 (complete bony fusion). Lower grades indicate more growth potential and greater risk of curve progression.2,3,14 At smaller degrees of curvature, radiographic monitoring for progression every six months is warranted, and referral is indicated if the angle increases2,14 (Table 17). Magnetic resonance imaging should be considered if alternative diagnoses such as malignancy are in the differential.
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