John Stockton holds the NBA record for career steals due to his skill in determining when he can steal the ball, not just his mechanical ability. This article presents an algorithm to determine if an object can intercept a moving ball based on their positions, velocities, and maximum speeds. It models the interception as occurring when their positions are equal at some time, derives the solution as a quadratic equation, and analyzes the possible cases - no solution, one solution, or two solutions - and their implications. It also discusses choosing the optimal intercept time and extending the model to three dimensions.
ROOT-LOCUS METHOD, Determine the root loci on the real axis /the asymptotes o...Waqas Afzal
Angle and Magnitude Conditions
Example of Root Locus
Steps
constructing a root-locus plot is to locate the open-loop poles and zeros in s-plane.
Determine the root loci on the real axis
Determine the asymptotes of the root loci
Determine the breakaway point.
Closed loop stability via root locus
This document describes the design of a robot system called Robo-Golf that can play golf autonomously. An overhead camera captures images of the golf field which are processed by a computer to determine the positions of the ball, hole, and robot. The robot uses vision and algorithms to navigate across the field and hit the ball into the hole using a golf club. It was tested over 17 trials and was able to successfully hit the ball into the hole 52.94% of the time on average in 1.24 shots. While successful, improvements are still needed in the image processing and navigation control systems.
This document discusses conformal mapping techniques for analyzing and optimizing quadrupole and sextupole magnet pole contours. It introduces conformal mapping as a tool to extend knowledge from dipole magnets to other magnet geometries. Equations are derived to relate optimized dipole pole cutoff dimensions to optimized quadrupole pole cutoff dimensions using conformal mapping. Graphs illustrate the field quality advantages of optimized poles. The document also discusses using conformal mapping to transform quadrupole and sextupole geometries into dipole geometries to improve the precision of magnetic field calculations.
This document describes Team NT's robotics team from Higashiyama High School in Kyoto, Japan. It discusses their 2021 robot design for the RoboCup Junior soccer competition, including omni-directional wheels, 3D printed parts, and software for path planning and ball tracking. It also summarizes their research on using hyperboloid mirrors for omni-vision sensors, which allows for accurate distance measurement and localization. This research earned them an award but is a novel approach not yet tried by other junior league teams.
ANÁLISIS SÍSMICO DE LOSA TRIANGULAR DE DOS PISOS.Rosbert Malpaso
ANÁLISIS SÍSMICO DE UNA EDIFICACIÓN DE LOSA TRIANGULAR DE DOS PISOS A NIVEL PREGRADO DESARROLLADO EN LA FACULTAD DE INGENIERÍA CIVIL DE LA UNASAM. (GRADOS DE LIBERTAD, MODOS DE VIBRAR, FRECUENCIAS, DESPLAZAMIENTOS, ETC).
This document contains a quiz with multiple choice questions about conic sections (circles, parabolas, hyperbolas, and ellipses). The quiz is divided into two parts. Part 1 contains 5 questions to identify which conic section (circle, parabola, hyperbola, or ellipse) equations are related to. Part 2 contains 10 word problems about conic sections, with multiple choice or short answer responses required. The document provides context, diagrams, and equations to help students answer the questions.
The document discusses relative motion using the example of a motorcyclist riding above flat cars moving on a train track. It states that the displacement and velocity of an object relative to another reference point is the sum of the displacements and velocities relative to each reference point. Specifically:
1) The displacement and velocity of the motorcyclist relative to the Earth is equal to the displacement and velocity relative to the flat cars plus the displacement and velocity of the flat cars relative to the Earth.
2) If the motorcyclist is moving at 13 m/s relative to the flat cars and the flat cars are moving at 30 m/s relative to the Earth, then the velocity of the motorcyclist relative to the
Surveillance refers to the task of observing a scene, often for lengthy periods in search of particular objects or particular behaviour. This task has many applications, foremost among them is security (monitoring for undesirable behaviour such as theft or vandalism), but increasing numbers of others in areas such as agriculture also exist. Historically, closed circuit TV (CCTV) surveillance has been mundane and labour Intensive, involving personnel scanning multiple screens, but the advent of reasonably priced fast hardware means that automatic surveillance is becoming a realistic task to attempt in real time. Several attempts at this are underway.
ROOT-LOCUS METHOD, Determine the root loci on the real axis /the asymptotes o...Waqas Afzal
Angle and Magnitude Conditions
Example of Root Locus
Steps
constructing a root-locus plot is to locate the open-loop poles and zeros in s-plane.
Determine the root loci on the real axis
Determine the asymptotes of the root loci
Determine the breakaway point.
Closed loop stability via root locus
This document describes the design of a robot system called Robo-Golf that can play golf autonomously. An overhead camera captures images of the golf field which are processed by a computer to determine the positions of the ball, hole, and robot. The robot uses vision and algorithms to navigate across the field and hit the ball into the hole using a golf club. It was tested over 17 trials and was able to successfully hit the ball into the hole 52.94% of the time on average in 1.24 shots. While successful, improvements are still needed in the image processing and navigation control systems.
This document discusses conformal mapping techniques for analyzing and optimizing quadrupole and sextupole magnet pole contours. It introduces conformal mapping as a tool to extend knowledge from dipole magnets to other magnet geometries. Equations are derived to relate optimized dipole pole cutoff dimensions to optimized quadrupole pole cutoff dimensions using conformal mapping. Graphs illustrate the field quality advantages of optimized poles. The document also discusses using conformal mapping to transform quadrupole and sextupole geometries into dipole geometries to improve the precision of magnetic field calculations.
This document describes Team NT's robotics team from Higashiyama High School in Kyoto, Japan. It discusses their 2021 robot design for the RoboCup Junior soccer competition, including omni-directional wheels, 3D printed parts, and software for path planning and ball tracking. It also summarizes their research on using hyperboloid mirrors for omni-vision sensors, which allows for accurate distance measurement and localization. This research earned them an award but is a novel approach not yet tried by other junior league teams.
ANÁLISIS SÍSMICO DE LOSA TRIANGULAR DE DOS PISOS.Rosbert Malpaso
ANÁLISIS SÍSMICO DE UNA EDIFICACIÓN DE LOSA TRIANGULAR DE DOS PISOS A NIVEL PREGRADO DESARROLLADO EN LA FACULTAD DE INGENIERÍA CIVIL DE LA UNASAM. (GRADOS DE LIBERTAD, MODOS DE VIBRAR, FRECUENCIAS, DESPLAZAMIENTOS, ETC).
This document contains a quiz with multiple choice questions about conic sections (circles, parabolas, hyperbolas, and ellipses). The quiz is divided into two parts. Part 1 contains 5 questions to identify which conic section (circle, parabola, hyperbola, or ellipse) equations are related to. Part 2 contains 10 word problems about conic sections, with multiple choice or short answer responses required. The document provides context, diagrams, and equations to help students answer the questions.
The document discusses relative motion using the example of a motorcyclist riding above flat cars moving on a train track. It states that the displacement and velocity of an object relative to another reference point is the sum of the displacements and velocities relative to each reference point. Specifically:
1) The displacement and velocity of the motorcyclist relative to the Earth is equal to the displacement and velocity relative to the flat cars plus the displacement and velocity of the flat cars relative to the Earth.
2) If the motorcyclist is moving at 13 m/s relative to the flat cars and the flat cars are moving at 30 m/s relative to the Earth, then the velocity of the motorcyclist relative to the
Surveillance refers to the task of observing a scene, often for lengthy periods in search of particular objects or particular behaviour. This task has many applications, foremost among them is security (monitoring for undesirable behaviour such as theft or vandalism), but increasing numbers of others in areas such as agriculture also exist. Historically, closed circuit TV (CCTV) surveillance has been mundane and labour Intensive, involving personnel scanning multiple screens, but the advent of reasonably priced fast hardware means that automatic surveillance is becoming a realistic task to attempt in real time. Several attempts at this are underway.
I am Martina J. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Maryland. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
This document provides a summary of key concepts in two-dimensional kinematics and projectile motion. It begins by defining displacement, velocity, and acceleration in two dimensions. It then discusses solving kinematics problems by resolving vectors into horizontal and vertical components. The document also covers projectile motion, where the horizontal velocity is constant and vertical acceleration is due to gravity. It ends by discussing relative velocity problems involving adding velocities of objects moving relative to each other or to a fixed point.
- Carlos Ragone fell 500 feet into snow after his anchor gave way while mountain climbing. The snow broke his fall, creating a 4 foot deep hole.
- Assuming constant acceleration during his impact with the snow, the estimated average acceleration was about 125g as he slowed to a stop within the 4 foot deep hole.
- A runner ran 2.5 km in 9 minutes, then walked back to the starting point over 30 minutes. Her average velocity was 5.2 km/hr for running, -0.83 km/hr for walking, and 0 km/hr for the total trip. Her average speed for the entire trip was 1.3 km/hr.
The document contains multiple physics problems related to rotation, torque, angular acceleration, and moments of inertia. One problem involves calculating the angular speed of a winch drum that is lifting a 2000 kg block at a constant speed of 8 cm/s. It is determined that:
- The tension in the cable equals 19.6 kN, given by the weight of the block
- The torque exerted on the winch drum by the cable is 5.9 mkN
- The angular speed of the winch drum is 0.27 rad/s
- The power required by the motor is 1.6 kW
This document introduces the polar coordinate system and provides examples of converting between polar and rectangular coordinates. It defines the polar coordinate system as having a point O as the pole, with a ray from O as the polar axis. Each point P is given by its distance r from O and the angle θ from the polar axis. The document gives conversion equations between polar (r, θ) and rectangular (x, y) coordinates and provides examples of plotting points, converting between systems, and rewriting equations in polar form.
COORDINATE GEOMETRY (7.4 Equations of Loci).pptxMuslimah1983
(i) The document discusses equations of loci, which are sets of points that satisfy certain geometric conditions. It provides examples of loci defined by a point being a constant distance from a fixed point or having a constant ratio of distances from two fixed points.
(ii) Key concepts covered include representing loci graphically, determining the equation of a locus by setting the condition equal to zero and simplifying, and solving problems involving loci.
(iii) Examples demonstrate finding equations of loci for different conditions like circles and ellipses, and proving that a point satisfies a locus equation.
College Physics 1st Edition Etkina Solutions ManualHowardRichsa
This document contains a chapter from the textbook "College Physics" by Etkina, Gentile, and Van Heuvelen. It includes multiple choice and conceptual questions about kinematics concepts like displacement, velocity, acceleration. It also includes practice problems asking students to draw motion diagrams and choose reference frames. The key concepts covered are scalar and vector quantities, relationships between displacement, velocity and acceleration, and using graphs to represent motion.
Point Placement Algorithms: An Experimental StudyCSCJournals
The point location problem is to determine the position of n distinct points on a line, up to translation and reflection by the fewest possible pairwise (adversarial) distance queries. In this paper we report on an experimental study of a number of deterministic point placement algorithms and an incremental randomized algorithm, with the goal of obtaining a greater insight into the practical utility of these algorithms, particularly of the randomized one.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
This document provides an overview of graphical analysis of motion in one dimension. It discusses key student difficulties in interpreting position-time, velocity-time, and acceleration-time graphs. It then presents examples of qualitative and quantitative analysis of constant velocity and accelerated motion. Sample problems from past AP Physics exams are also included to illustrate the application of graphical analysis techniques.
The document provides information about a JEE Advanced exam paper from 2020. It contains 6 multiple choice questions in Section 1 of the Physics part of the exam paper. Section 2 contains 6 additional questions, some with multiple correct answers. The questions cover topics in mechanics, electromagnetism, optics, and thermodynamics. Key concepts assessed include circular motion, magnetic fields, refraction of light, fluid statics, Bohr model of the hydrogen atom, blackbody radiation, and dimensional analysis.
This document discusses parametric equations and motion. It defines parametric curves as graphs of ordered pairs (x,y) where x and y are functions of a parameter t. Parametric equations can be used to model the path of objects in motion. Examples are provided of graphing parametric equations, eliminating the parameter to get a standard equation, and finding parametric equations to represent a line. The document also contains sample problems testing understanding of parametric equations and related concepts.
This document reports on a project to model a serial and parallel manipulator. The serial manipulator is a PPRR configuration that can trace an ellipse through a GUI. The parallel manipulator uses a 4RRR configuration with rhombus links to trace trajectories by adjusting link lengths and angles. Implementing null space control improved the parallel manipulator's ability to follow trajectories despite parameter changes. Both manipulators provided insight into applying theoretical robotic concepts to practical problems.
Calculating the Angle between Projections of Vectors via Geometric (Clifford)...James Smith
We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Signatures of wave erosion in Titan’s coastsSérgio Sacani
The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.
Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...Creative-Biolabs
Neutralizing antibodies, pivotal in immune defense, specifically bind and inhibit viral pathogens, thereby playing a crucial role in protecting against and mitigating infectious diseases. In this slide, we will introduce what antibodies and neutralizing antibodies are, the production and regulation of neutralizing antibodies, their mechanisms of action, classification and applications, as well as the challenges they face.
I am Martina J. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Maryland. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
This document provides a summary of key concepts in two-dimensional kinematics and projectile motion. It begins by defining displacement, velocity, and acceleration in two dimensions. It then discusses solving kinematics problems by resolving vectors into horizontal and vertical components. The document also covers projectile motion, where the horizontal velocity is constant and vertical acceleration is due to gravity. It ends by discussing relative velocity problems involving adding velocities of objects moving relative to each other or to a fixed point.
- Carlos Ragone fell 500 feet into snow after his anchor gave way while mountain climbing. The snow broke his fall, creating a 4 foot deep hole.
- Assuming constant acceleration during his impact with the snow, the estimated average acceleration was about 125g as he slowed to a stop within the 4 foot deep hole.
- A runner ran 2.5 km in 9 minutes, then walked back to the starting point over 30 minutes. Her average velocity was 5.2 km/hr for running, -0.83 km/hr for walking, and 0 km/hr for the total trip. Her average speed for the entire trip was 1.3 km/hr.
The document contains multiple physics problems related to rotation, torque, angular acceleration, and moments of inertia. One problem involves calculating the angular speed of a winch drum that is lifting a 2000 kg block at a constant speed of 8 cm/s. It is determined that:
- The tension in the cable equals 19.6 kN, given by the weight of the block
- The torque exerted on the winch drum by the cable is 5.9 mkN
- The angular speed of the winch drum is 0.27 rad/s
- The power required by the motor is 1.6 kW
This document introduces the polar coordinate system and provides examples of converting between polar and rectangular coordinates. It defines the polar coordinate system as having a point O as the pole, with a ray from O as the polar axis. Each point P is given by its distance r from O and the angle θ from the polar axis. The document gives conversion equations between polar (r, θ) and rectangular (x, y) coordinates and provides examples of plotting points, converting between systems, and rewriting equations in polar form.
COORDINATE GEOMETRY (7.4 Equations of Loci).pptxMuslimah1983
(i) The document discusses equations of loci, which are sets of points that satisfy certain geometric conditions. It provides examples of loci defined by a point being a constant distance from a fixed point or having a constant ratio of distances from two fixed points.
(ii) Key concepts covered include representing loci graphically, determining the equation of a locus by setting the condition equal to zero and simplifying, and solving problems involving loci.
(iii) Examples demonstrate finding equations of loci for different conditions like circles and ellipses, and proving that a point satisfies a locus equation.
College Physics 1st Edition Etkina Solutions ManualHowardRichsa
This document contains a chapter from the textbook "College Physics" by Etkina, Gentile, and Van Heuvelen. It includes multiple choice and conceptual questions about kinematics concepts like displacement, velocity, acceleration. It also includes practice problems asking students to draw motion diagrams and choose reference frames. The key concepts covered are scalar and vector quantities, relationships between displacement, velocity and acceleration, and using graphs to represent motion.
Point Placement Algorithms: An Experimental StudyCSCJournals
The point location problem is to determine the position of n distinct points on a line, up to translation and reflection by the fewest possible pairwise (adversarial) distance queries. In this paper we report on an experimental study of a number of deterministic point placement algorithms and an incremental randomized algorithm, with the goal of obtaining a greater insight into the practical utility of these algorithms, particularly of the randomized one.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
This document provides an overview of graphical analysis of motion in one dimension. It discusses key student difficulties in interpreting position-time, velocity-time, and acceleration-time graphs. It then presents examples of qualitative and quantitative analysis of constant velocity and accelerated motion. Sample problems from past AP Physics exams are also included to illustrate the application of graphical analysis techniques.
The document provides information about a JEE Advanced exam paper from 2020. It contains 6 multiple choice questions in Section 1 of the Physics part of the exam paper. Section 2 contains 6 additional questions, some with multiple correct answers. The questions cover topics in mechanics, electromagnetism, optics, and thermodynamics. Key concepts assessed include circular motion, magnetic fields, refraction of light, fluid statics, Bohr model of the hydrogen atom, blackbody radiation, and dimensional analysis.
This document discusses parametric equations and motion. It defines parametric curves as graphs of ordered pairs (x,y) where x and y are functions of a parameter t. Parametric equations can be used to model the path of objects in motion. Examples are provided of graphing parametric equations, eliminating the parameter to get a standard equation, and finding parametric equations to represent a line. The document also contains sample problems testing understanding of parametric equations and related concepts.
This document reports on a project to model a serial and parallel manipulator. The serial manipulator is a PPRR configuration that can trace an ellipse through a GUI. The parallel manipulator uses a 4RRR configuration with rhombus links to trace trajectories by adjusting link lengths and angles. Implementing null space control improved the parallel manipulator's ability to follow trajectories despite parameter changes. Both manipulators provided insight into applying theoretical robotic concepts to practical problems.
Calculating the Angle between Projections of Vectors via Geometric (Clifford)...James Smith
We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Signatures of wave erosion in Titan’s coastsSérgio Sacani
The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.
Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...Creative-Biolabs
Neutralizing antibodies, pivotal in immune defense, specifically bind and inhibit viral pathogens, thereby playing a crucial role in protecting against and mitigating infectious diseases. In this slide, we will introduce what antibodies and neutralizing antibodies are, the production and regulation of neutralizing antibodies, their mechanisms of action, classification and applications, as well as the challenges they face.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Candidate young stellar objects in the S-cluster: Kinematic analysis of a sub...Sérgio Sacani
Context. The observation of several L-band emission sources in the S cluster has led to a rich discussion of their nature. However, a definitive answer to the classification of the dusty objects requires an explanation for the detection of compact Doppler-shifted Brγ emission. The ionized hydrogen in combination with the observation of mid-infrared L-band continuum emission suggests that most of these sources are embedded in a dusty envelope. These embedded sources are part of the S-cluster, and their relationship to the S-stars is still under debate. To date, the question of the origin of these two populations has been vague, although all explanations favor migration processes for the individual cluster members. Aims. This work revisits the S-cluster and its dusty members orbiting the supermassive black hole SgrA* on bound Keplerian orbits from a kinematic perspective. The aim is to explore the Keplerian parameters for patterns that might imply a nonrandom distribution of the sample. Additionally, various analytical aspects are considered to address the nature of the dusty sources. Methods. Based on the photometric analysis, we estimated the individual H−K and K−L colors for the source sample and compared the results to known cluster members. The classification revealed a noticeable contrast between the S-stars and the dusty sources. To fit the flux-density distribution, we utilized the radiative transfer code HYPERION and implemented a young stellar object Class I model. We obtained the position angle from the Keplerian fit results; additionally, we analyzed the distribution of the inclinations and the longitudes of the ascending node. Results. The colors of the dusty sources suggest a stellar nature consistent with the spectral energy distribution in the near and midinfrared domains. Furthermore, the evaporation timescales of dusty and gaseous clumps in the vicinity of SgrA* are much shorter ( 2yr) than the epochs covered by the observations (≈15yr). In addition to the strong evidence for the stellar classification of the D-sources, we also find a clear disk-like pattern following the arrangements of S-stars proposed in the literature. Furthermore, we find a global intrinsic inclination for all dusty sources of 60 ± 20◦, implying a common formation process. Conclusions. The pattern of the dusty sources manifested in the distribution of the position angles, inclinations, and longitudes of the ascending node strongly suggests two different scenarios: the main-sequence stars and the dusty stellar S-cluster sources share a common formation history or migrated with a similar formation channel in the vicinity of SgrA*. Alternatively, the gravitational influence of SgrA* in combination with a massive perturber, such as a putative intermediate mass black hole in the IRS 13 cluster, forces the dusty objects and S-stars to follow a particular orbital arrangement. Key words. stars: black holes– stars: formation– Galaxy: center– galaxies: star formation
HUMAN EYE By-R.M Class 10 phy best digital notes.pdf
intercepting a ball.pptx
1. 9.8
Intercepting a Ball
Noah Stein
noah@acm.org
J ohn Stockton holds the NBA all-time record for the most steals by an individual
throughout his career-more than 2,800 (as of2001). He has more than luck. He
has skill. However, his skill is not merely confined to the mechanical act of stealing
the ball. As imponant as the motion itself, he has skill in determining when he can
steal the ball. Without the when, he would never have the chance to use the how.
In sports games, many situations arise that require determining whether a pl ay er
can intercept the ball: a second baseman wants to catch a line drive, a hockey wing
tries to steal a pass, a soccer goalie needs to block a shot, or a basketball center wishes
to rebound the ball. In these cases, the AI needs to decide if the action can be success
fully executed, or whether an alternative course of action should be taken.
The algorithm described herein is also applicable outside the genre of sports
games: the Shao-Lin master might need to decide whether to grab an arrow flying at
him or just move his leg out of the way. Can the Patriot missile intercept and destroy
the Scud? A secret service agent needs to know if he can dive in front of the President
and take the bullet in time.
The Basic Problem
The previous examples can be distilled down to essentially the same problem: one
object is at a certain position (P b ) and traveling in a straight line at a constant velocity
(Vb); another object at a different position (P p ) wants to intercept the first, but it
might not move faster than a specific speed (s). From this information, the model
solves for the second object's velocity to intercept (i,;,). Some might object to the sim
plicity of the model; however, every model must make simplifications, and those
made for this problem render a functional solution. Please note that in the discussion
that follows, the ball is the object that is traveling along a path that is to be inter
cepted, and the player is the object that wants to intercept the ball.
The first simplification is that the ball's velocity is constant and its trajectory is
therefore a straight line. A basketball coming off the rim follows a parabolic trajec
tory-not even remotely close to a straight line. How can a straight line model the
motion effectively? This model can be broken down into two independent submod
els: the altitude of the ball, and the motion in the ground plane. Because the two axes
2. 496 Section 9 Racing and Sports Al
in the ground plane are orthogonal to the altitude axis, their motions can be consid
ered in isolation [Resnick91]. The bulk of this article explains the computation of the
ground plane interception. The end of the article addresses adding altitude consider
ations to the model.
The second simplification is that the intercepting object has no turning radius,
infinite acceleration, and can travel indefinitely at its maximum velocity. This is defi
nitely the more difficult of the two aspects to explain, because it will introduce the
most error. First, error isn't necessarily a bad thing: a real person will have difficulty
judging certain situations. Second, there are other methods that can be used to com
pensate for heading changes, some of which appear later. In addition, the interplay
between changes of direction and acceleration is so complex that a simplification to
some degree must occur.
Given this information, the model has four independent variables: the position of
the ball, the velocity of the ball, the position of the interceptor, and the maximum
speed the interceptor can travel. A graphical representation of the problem is shown
in Figure 9 .8.1. Please note that in all figures, dots represent the objects, solid lines
represent velocities, and dashed lines represent changes in position.
O ······
➔
? A ?
't: : -:1
••
t=0
0------ ➔
? A ?
�. : ..�
·-
....•....
··
t = 1
FIGURE 9.8.1 The intercept ball problem.
O ······➔
�
? ?
�--. • .,. -:1
·--.........
·····
Deriving the Solution
Examining Figure 9.8.1, the problem might appear to be the point on line closest to
point algorithm from Graphics Gems [Glassner90]. The check determines the closest
point on the trajectory line. It appears to be a good choice. In some cases, the optimal
choice is close to this point; however, Figure 9.8.2 clearly illustrates a case in which
the point-line test clearly and convincingly fails to deliver the correct result.
What is the proper mathematical model? For an interception to occur, the posi
tion of both the ball and the player must be the same at some time t. Thus, if V P was
known a priori, the intercept statement would appear as such:
P b + V b t == P p + vl (9.8.1)
3. 9,8 Intercepting a Ball 497
O· ·····➔ O ······
➔
FIGURE 9.8.2 Failure to intercept the ball given the simple solution of choosing the closest
point on the ball's trajectory.
Unfortunately, V P is the variable to be solved. The solution requires that the
problem be viewed from a different vantage. If the positions of the player and ball are
the same, the distance must be 0. The distance between the player's initial position
and the ball at time r.
(9.8.2)
In Equation 9.8.2, the vector can be considered to be composed of two elements:
the initial position of the ball relative to the player (the part in the parentheses), and
the motion of the ball due to its velocity vector. If the player can move a distance
equivalent to how distant the ball is, the player can intercept the ball at time t:
(9.8.3)
Since the initial positions never change, to simplify further discussion, the substi
tution P =P b - P; will be made from this point forth. In addition, subscripts will be
dropped since there will be no ambiguity in the text. Solving for t results in the
time(s) at which the player can successfully intercept the ball:
IP+ Vt != st
.J(P +Vt}• (P + Vt} = st
(P +Vt}• (P +Vt)= (st)
2
P • P + 2P • Vt+ V • Vt 2
= s2
t 2
(V • V - s
2
)t 2 + ( 2P • V )t + ( P • P) = 0 (9.8.4)
Now the equation is a second-order polynomial oft. Plug the polynomial scalars
into the quadratic equation.
4. 498 Section 9 Racing and Sports Al
-----
Analysis of the Quadratic
- b ± ✓
b 2 - 4ac
2a
(9.8 .5)
The quadratic equation has three different categories of solutions: no real roots,
one real root, and two real roots. The category of solution is determined by the
expression in the radical: b2 -4ac. If it is negative, the solution has no real roots. If it
is zero, the radical after the "±" is zero and results in a single real root. If greater than
zero, the solution has two real roots. But what does this mean? First, let's transform
the expression in the radical into a more informative form for subsequent analysis:
b 2 - 4 ac = (2P • v) 2
- 4 (v • v - s2
)(r • r)
= 4(P • V
r -4(V • V - s2
)( P • P)
= (P • vr -(v • V - s2
)(r • P )
= (r • v) 2
+ (s2 - v • v )(r • r )
No Real Roots
(9. 8 .6 )
If the radicand (the quantity within the radical ) is negative, then there are no real
roots, and the ball cannot be intercepted. In this case, the quantity (s 2 - V • V) must
be negative, so s</V/. Only when the ball travels at a speed greater than the maximum
speed of the player will chis case occur. This agrees with our intuition that the player
has co be able to move faster than the ball if he ever hopes co intercept its path.
One Real Root
This case represents a border case between whether or not the player can intercept
the ball. The interception is so difficult that there is only one point in time chat the
ball can be intercepted. For the quadratic equation to result in a single root, the radi
cand must be zero. Examining Equation 9.8.6, if the initial positions coincide, the
radical is zero, because both addends have multiplicands that have doc products
involving P, resulting in zeros. Equation 9.8.5 reduces to:
- b - (P • v )
2 a 2(v • v - s2
)
To fully understand the single real root case, Equation 9.8.7 must be considered
in light of the face that the radicand is zero. From Equation 9.8.6:
5. 9.8 Intercepting a Ball 499
(P • v)
2
+ (s2
- V • v )(P • P) = 0
( P • V )
2
= - ( s2
- V • V )(P • P) (9.8.8)
Since the left side must be positive (due to the square), the right side must be neg
ative; therefore, the interceptor is faster than the ball. Examining Equation 9.8.7 with
that knowledge, the divisor must be negative in this case. Our analysis of the single
real root has two subcases:
• (P•V)<O: The ball's velocity is roughly toward the interceptor, thus it can be inter
cepted at some point in the future. The numerator becomes negative (because of
the negative sign), so the equation has a positive result. Since the ball moves faster
than the interceptor, it can only be intercepted at one point in time. After that time,
it will be traveling too quickly to be reached again. This is akin to the second base
man grabbing a line drive: if it shoots by in arm's reach, he can grab it in his imme
diate vicinity, but he'll never have a chance to catch it in the outfield.
• (P•V)>O: The ball's velocity is not toward the interceptor in any conceivable way.
So, how is it that there is a root? With the dividend negative, the result is nega
tive. Thus, the interception occurred in the past. Since time only moves forward,
this result indicates that the result should be discarded.
Two Real Roots
The final case is the mo st interesting. This case does not require the speed of the
pl ay er to be greater. Although the ball can move significantly faster than the player, if
the player is sufficiently far away from the ball's initial position, but near the line in
positive t, the ball can be intercepted.
The two real roots represent the boundaries of a window of opportunity; how
ever, their interpretation falls into three categories, depending on the sign of the roots:
two positive roots, two negative roots, and one positive and one negative root.
• Two p ositive roots: The roots define a window of opportunity in which the ball
can be successfully intercepted. Any time between the two roots is a valid time at
which the ball can be intercepted. In this scenario, the ball is moving faster than
the pl ay er, but the player is, relative to the ratio of speed and distance, close to the
line of motion and thus can make it there in time.
• Two negative roots: The roots also define a window of opportunity between the
two values in which the ball can be intercepted. It also has the property of the ball
moving faster than the pl ay er; however, the ball is moving entirely away from the
player. Thus, the window is entirely in negative time, so this result is to be dis
carded as an impossible interception.
• One p ositive and one negative root: The solution has two open-ended windows
of opportunity: time less than or equal to the negative root, and time greater than
6. 500 Section 9 Racing and Sports Al
is to be discarded. In this scenario, the player is moving faster than the ball, and
thus can reach at any time after a certain minimum needed to catch it.
Choosing the Time to Meet
Once the root or roots are known, a valid time can be plugged back into the first
equation in Equation 9.8.4, resulting in the point at which the target can be inter
cepted. Of the three root categories, only the two real roots case affords the AI discre
tion in choosing at which time the player would like to intercept the ball. In the no
real roots case, the ball cannot be intercepted, and with a single real root, there is a
unique time at which the interception could occur.
The two real roots case, in contrast, defines an interval of time in which the ball
can be intercepted. What is the best time to choose? The obvious answer is a root
itsel£ Although the only answer for the single real root case, it is probably not the best
choice in the two-root case. To illustrate, imagine one person passing the ball to
another. T he passer throws the ball just beyond the receiver's reach. The receiver
could take two leisurely steps perpendicular to the path of the ball and grab it. If a
root is chosen, he will run as fast as he can to catch the ball, running mostly in a direc
tion parallel to the motion of the ball.
Thus, for a more realistic decision, a time somewhere in the middle of interval
should be chosen. T ry to find the "lazy" point-the solution requiring the least
amount of effort to still create an interception outcome. How can the lazy point be
determined? We know two aspects of the solution: 1) the speed of interception should
be as small as possible, yet still allow an interception to occur, and 2) at the minimum
allowed interception speed, there is only one possible point in time to intercept, so
there must be only one real root. For one real root, the expression under the radical
must be zero. Solve for s:
(P • V
r+ ( s
2
- V • V ) (P • P) = O
(P • v f + s
2
(P • P)- (v • v) (P • P) = 0
2 _ ( V • V )( P • P) - ( P • V
r
s
s =
(P • P)
(v • v)( P • P )- ( P • v f
(P • P)
(9.8.9)
7. 9.8 Intercepting a Ball 501
The solution for s in Equation 9.8.9 must then be plugged back into the qua
dratic equation to give the resulting time of interception that can then be used to
compute the location of intersection.
Related Problems
Now that the essential model has been fully constructed and anal yz ed, let's briefly
summarize two variants:
1. The pl ay er has long arms that the model should consider. T he arms can be
modeled effectively as a nonzero initial position. If the player's arms are l
long, Equation 9.8.3 becomes:
l(Pb - P;) + V b t l = l + s;t (9.8.10)
The new solutions can be derived as above from this new equation.
2. Another consideration that might require modeling could be a del ay ed reac
tion by the player. Some developers' preference is to wait until such time as
the player can move on the ball to make the decision; however, your needs
might require a predetermination. As such, if the player has a delay of d
duration, Equation 9.8.3 becomes:
(9.8.11)
These two variants can be used together in a single statement by merely
replacing the t factor in Equation 9.8.10 with the (t-d} factor seen in Equa
tion 9.8.11.
Rebounding the Ball
In rebounding, the ball's altitude affects a player's ability to intercept a ball. As indi
cated at the beginning of the article, the model thus far only handles the relationship
between the ball and player in the ground plane. In rebounding and similar situa
tions, a second time window is computed. In general, the intersection of two parabo
las has four solutions; however, the alignment of the parabolas results in a single
i ntersection, with the exception of the case where the ball and player coincide. The
formulation of the equation is reminiscent of the planar check-if the altitude of
the player is greater than the ball, he can intercept it.
P + s t - 1/ gt 2
> p + s t - ½'gt 2
p p / 2 h b
2
(9.8.12)
8. 502 Section 9 Racing and Sports Al
At time t greater than the right-hand side, the ball can be caught. In many situa
tions, P p must factor in the player's height and reach for acceptable results.
Once the altitude window has been determined, use the time value to trim down
the range returned by the plane check. Another modeling note: the interaction of ball
and backboard results in many motion discontinuities. The proper method of han
dling this is to perform one check for each region of continuous time.
Conclusion
This article presents a simple and concise method to determine if one object can be
intercepted by another. The check consists of little more than three dot products to
determine the coefficients of a quadratic equation. The value of the expression under
the radical in the quadratic equation discriminates between the three major cases:
interception is impossible, a single point in time to intercept, and a window of oppor
tunity for interception. If a window of opportunity is found, further analysis deter
mines the best choice in the window.
The algorithm operates on a much-simplified view of the game model. The sim
plified model increases the error of the check; however, the error can be reduced effec
tively by dividing up the problem space and running each parameterization through
the algorithm. The method's efficiency confers the advantage that it can be run
frequently.
References
[Anton00] Anton, Howard, Elementary Linear Algebra, 8 th
Ed., John Wiley & Sons,
2000.
[Glassner90] Glassner, Andrew, "Useful 2D Geometry," Graphics Gems, Academic
Press, 1990.
[Resnick91] Resnick, Robert, and Halliday, David, Physics, 4 th
Ed., Vol 1, John Wiley
& Sons, 1991.
[Spiegel97] Spiegel, Murray R., and Rabin, Schaum's Outline of College Algebra, 2 nd
Ed., McGraw-Hill Professional Publishing, 1997.