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Borwein, J. New York: Wiley, Gray, A. Kabai, S. Lawrence, J. A Catalog of Special Plane Curves. New York: Dover, pp. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. Lockwood, E. A Book of Curves. Cambridge, England: Cambridge University Press, MacTutor History of Mathematics Archive. Sloane, N. Smith, D. History of Mathematics, Vol. New York: Dover, p. Wells, D. London: Penguin, pp.

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Note the symbols used for field into and out of the paper. Mapping Magnetic Field Lines : Small compasses could be used to map the fields shown here. A The magnetic field of a circular current loop is similar to that of a bar magnet. B A long and straight wire creates a field with magnetic field lines forming circular loops. C When the wire is in the plane of the paper, the field is perpendicular to the paper. Note that the symbols used for the field pointing inward like the tail of an arrow and the field pointing outward like the tip of an arrow.

Extensive exploration of magnetic fields has revealed a number of hard-and-fast rules. We use magnetic field lines to represent the field the lines are a pictorial tool, not a physical entity in and of themselves.

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The properties of magnetic field lines can be summarized by these rules:. The last property is related to the fact that the north and south poles cannot be separated. It is a distinct difference from electric field lines, which begin and end on the positive and negative charges. If magnetic monopoles existed, then magnetic field lines would begin and end on them. Earth is largely protected from the solar wind, a stream of energetic charged particles emanating from the sun, by its magnetic field, which deflects most of the charged particles.

These particles would strip away the ozone layer, which protects Earth from harmful ultraviolet rays. The region above the ionosphere, and extending several tens of thousands of kilometers into space, is called the magnetosphere. This region protects Earth from cosmic rays that would strip away the upper atmosphere, including the ozone layer that protects our planet from harmful ultraviolet radiation. The magnetic field strength ranges from approximately 25 to 65 microteslas 0. The intensity of the field is greatest near the poles and weaker near the equator.

These effects can be combined into a partial differential equation called the magnetic induction equation:. In this equation u is the velocity of the fluid, B is the magnetic field, and eta is the magnetic diffusivity. The first term on the right hand side of the induction equation is a diffusion term. The motion of the molten outer iron core is sustained by convection, or motion driven by buoyancy. The temperature increases toward the center of Earth, and the higher temperature of the fluid lower down makes it buoyant.

The Coriolis effect, caused by the overall planetary rotation, tends to organize the flow into rolls aligned along the north-south polar axis. Electric currents induced in the ionosphere generate magnetic fields ionospheric dynamo region. Such a field is always generated near where the atmosphere is closest to the sun, causing daily alterations that can deflect surface magnetic fields by as much as one degree.

Typical daily variations of field strength are about 25 nanoteslas nT , with variations over a few seconds of typically around 1 nT. The geomagnetic field changes on time scales from milliseconds to millions of years. Shorter time scales mostly arise from currents in the ionosphere ionospheric dynamo region and magnetosphere, and some changes can be traced to geomagnetic storms or daily variations in currents.

At present, the overall geomagnetic field is becoming weaker; the present strong deterioration corresponds to a 10 to 15 percent decline over the last years and has accelerated in the past several years. Geomagnetic intensity has declined almost continuously from a maximum 35 percent above the modern value achieved approximately 2, years ago. These events are called geomagnetic reversals.


Evidence for these events can be found worldwide in basalts, sediment cores taken from the ocean floors, and seafloor magnetic anomalies. Reversals occur at apparently random intervals ranging from less than 0. The most recent such event, called the Brunhes-Matuyama reversal, occurred about , years ago. Skip to main content. Search for:. Magnetism and Magnetic Fields Electric Currents and Magnetic Fields An electric current will produce a magnetic field, which can be visualized as a series of circular field lines around a wire segment.

Learning Objectives Describe shape of a magnetic field produced by an electric current flowing through a wire. Key Takeaways Key Points A wire carrying electric current will produce a magnetic field with closed field lines surrounding the wire. Another version of the right hand rules can be used to determine the magnetic field direction from a current—point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it.

Calculate the deflection angle intersection angle Cord Length & area of Curve

The Biot-Savart Law can be used to determine the magnetic field strength from a current segment. A current-carrying wire feels a force in the presence of an external magnetic field. Key Terms Biot-Savart Law : An equation that describes the magnetic field generated by an electric current.

It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. Permanent Magnets Permanent magnets are objects made from ferromagnetic material that produce a persistent magnetic field. Learning Objectives Give examples and counterexamples of permanent magnets. Key Takeaways Key Points Permanent magnets are objects made from magnetized material and produce continual magnetic fields. Moreover, all the participants have rich experience in mountain roads driving because two-lane mountain roads are very common in Chongqing, China.

Although their familiarity with the specified roads were not as good as that of local drivers. In this study, the vehicle trajectory is defined by the movement of the vehicle centroid. The positional relationships between the vehicle and the pavement markers such as the centerline and edge line for different ranges of the LDRT is list in Table 2. Obtaining continuous vehicle trajectory and determining the lateral deviation rate of trajectory LDRT.

Using the position of right tire on the pavement to determine the lateral position of the test car. The trend of LDRT curve depending on the travelled time before the first peak point, rise or decline;. Geometric parameters of the curves used in Section 3. The running time on the different curves varied owing to the differing curve geometries, therefore the different LDRT profiles were interlaced and the feature points were staggered, which made it difficult to determine the overall variation tendency and common control laws. Then they were translated and stretched to obtain a clearer trend and characteristics of LDRT profiles of bends with different geometric features, as shown in Fig.

Secondly, the LDRT values after entering the curves initially decrease monotonically to the minimum value before the mid-curve point approximately at the middle of the distance along the curve. Thus, the apexes of the bends could be considered to be located before the mid-curve point. Thirdly, the LDRT values gradually increase after C 1 , reaching the maximum value at P 1 before the end of the curve, indicating that the vehicles had moved from the left to the right side of the lane.

Fourthly, the LDRT values tend to decrease again after P 1 , in what is referred to as the recovery phase, in which the status before entering the curve is restored. Figure 6 a4 and a5 show two typical shapes of this trajectory pattern, and the point C 1 and P 1 were marked on this bend to give a visual presentation for an easier understanding.

In general, a too close distance between vehicle and road edge at point P 1 is more likely to occur when a higher speed at curve entry adopt by the driver. An outstanding feature of this driving pattern is that the cutting point C 1 appears on the middle of the bend, and P 1 no longer occurs within curve areas. The third common pattern of vehicle track is presented in Fig. The last track pattern observed on left bends is provided in Fig.

By this way, drivers can gain a minimum travel time throughout the bend, and this driving pattern is often seen on sharp bends with large deflection angle.

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Of the four patterns above, Pattern I and Pattern III are more frequently observed on younger drivers or unskilled drivers who have less driving experience, they might have an inaccurate perception of bend curvature and length, and thus choose an inappropriate speed and path.

For the perspective of traffic safety and collision prevention, the behavior of lane departures can be classified into two categories: active departure and passive departure. Whilst the latter is an unintended departure owing to a larger inertia of the quickly moving vehicle, that is, the vehicle moved towards the outside of the curve after curve-cutting is finished, and the departure at P 1 of track pattern I falls into this category. Pattern III is considered another dangerous track type, because the encroachment of opposite lane after C 1 can be grouped into passive departure.

Although the risk of heading on with the car driving on the opposite lane may exist in the active departures, such as near the point C 1. However, the risk of scraping or heading on the barrier mounted on the right side at point P 1 because of passive lane departure needs more immediate attention. With the Pattern II in Fig. However there exists a crucial distinction, i. The track shape of Pattern V is markedly distinct from the tracks of Pattern I to Pattern IV, drivers steered their cars toward the outside near the mid-point of the bend after they enter the bend.

With this pattern, drivers may get a better sight distance at curve centre, however, a risk of collision with opposite traffics is correspondingly brought out to drivers. This pattern, along with Pattern I, are more common in drivers who have less driving experience.

From the aspect of collision possibility at the position near P 1 for track Pattern I, IV and V, car body would be beyond the centerline of the roadway, and on the surface, which might have caused a collision with the vehicle running on the opposite lane. In particularly, the locally highest curvatures generally occurred around P 1 with track Pattern V because of the contribution of the steering corrections, and the increased lateral acceleration of this track type can cause the car rollover or sideslip.

Of the five presented track patterns in Fig. Using this criteria, Pattern II in Fig. There exists a connection between speed and trajectory shapes. Therefore, from this perspective, reducing the vehicle speed before it entering the curve entry is an effective measure to regularize the track behavior of passenger cars. LDRT profiles: a driver 1, left bends; b driver 4, left bends; c driver 7, left bends; d driver 1, right bends; e driver 4, right bends; f driver 7, right bends; g clustering of LDRT profiles based on the curvature radius.

It can be observed from Fig. In addition, when a vehicle enters a right bend with a small radius see Fig. Trajectory at a sharp left bend and b large-radius left bend; c — e typical trajectories at small-radius right bend. Safety countermeasures for preventing passive lane departures or inhibiting the behavior of curve cutting.

The factors related to road geometric parameters analyzed in this study are deflection angle, deflection direction and radius of horizontal alignment. And a greater the mass of the vehicle is, a bigger impact will be caused. In our observation from the videos recorded by the camera mounted on the front window, drivers are willing to cut curves if no oncoming vehicles on the opposite lanes regardless of whether driving on an uphill or downhill. Take downhill for instance, there are no indications that the slope enable the drivers to steer their vehicle in the middle of the driving width.

Therefore, slopes of mountain roads can be considered as an insignificant factor that can affect the track of passenger cars. Driving habits were analyzed under conditions in which the driver was free to choose his trajectory and driving width on the entire pavement, i. Data corresponding to scenarios such as vehicle following, overtaking, and curve meeting meeting a car in the oncoming lane at a curve were excluded from the analysis, and therefore, the observations and conclusions of this paper are based on data obtained during free-flow driving.

The exclusion was for obvious reasons. The trajectory of a vehicle is the output of a complex movement mechanism steered by a driver. Thus, using a given vehicle model, different drivers and driving behaviors input will definitely produce different trajectory patterns output.

Therefore, drivers with a wide range of age and driving experience were employed in this study to obtain as many trajectory patterns for two-lane mountain roads as possible. For the same reason, four roads with similar conditions were used to perform the driving test, so that we can extract more driving patterns from naturalistic driving data.


Our work are mainly intended to investigate the trajectory patterns on two-lane mountain roads, and focused on the effects of the geometrical factors of the road on the trajectory characteristics. The driver characteristics such as age and driving experience were not discussed, and these factors will be considered in another paper by the authors.

Different vehicles of two types were used in this work, including SUV and MPV, the reason for vehicle variability is that the selected MPVs are with seven seats, longer wheel base and heavier unladen mass, which resulting a relatively poor maneuverability and lower travelling speed than SUVs and other passenger car types. Therefore, for road infrastructures design and accident prevention it is a typical vehicle type. And we imagine if MPVs with larger size exhibit a diversity in track patterns, other passenger cars will truly reflect it.

One of limitations of the present study is that the experimental data were obtained from only male drivers. The driving behaviors of female drivers such as the speed choice, path selection and operation input on the test roads analyzed in this study which included numerous sharp curves and curved sloping sections might be different from that of male drivers, however, we need field test data to verify it. In a future work, we would perform experiments using real cars and female participants.

Another limitation of this study is that the sample of participants for each test road perhaps was insufficient, which could result in a consequence that some unknown track patterns may be missed. It should also be noted that large vehicles such as buses, coaches, and heavy trucks account for a large proportion of road vehicles, and their spatial trajectory is a main consideration in the design of road geometry.

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The trajectories of such large vehicles can only be accurately described by multiple feature points owing to their large bodies and long wheelbases. We plan to conduct further study using such vehicles. Typical track patterns within curve areas were determined according to features of LDRT profiles, four patterns for left-hand bends and five patterns for right-hand bends. Crash prone position and its corresponding accident types were analyzed.

The position of the apex cutting point is not fixed for different track patterns, and it may appear at the front of, just in and at the back of the middle of a curve. With increasing curve radius, the LDRT fluctuation weakens and the trajectory fits better with the horizontal alignments of the road. The smaller the radius, the more frequently the vehicle occupies the oncoming lane. The encroachment on the oncoming lane occurs when the vehicle enters a left bend or exits a right bend.

A vehicle encroaches the oncoming lane on two occasions, namely, during curve cutting and overcorrection. In the second case, the vehicles approaches too close to the outer side of the curve after cutting, and the drivers has to steer back to correct the lateral location of the vehicle. In other words, a vehicle approaches the oncoming lane due to poor steering of the driver. Based on the feature points within curve areas, the behavior of lane departures was classified into active departure and passive departure, of which passive lane departure might be at a greater risk of collision with roadside barrier and opposite traffic, and requires safety countermeasures to regularize the track.

The findings above contribute to judge the safety implications of driver behavior, and thus identify crash prone positioning and the potential mechanisms of head-on crashes, run-off-road and guardrail collisions. And therefore, effective countermeasures in safety facilities, speed enforcement and driver training could be acquired to raise the road safety.

Moreover, our findings may also be useful to automatic drive, which can provide helpful insight into the track behavior of passenger cars and track patterns recognition, and they could be useful in developing an anthropomorphic algorithm to simulate the driving habits of human drivers. The ultimate goal is to produce a human-like self-driving vehicle. Skip to main content Skip to sections.

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