To use the reference angle calculator, simply enter any angle into the angle box to find its reference angle, which is the acute angle that corresponds to the angle entered. The calculator automatically applies the rules we’ll review below.
Imagine a coordinate plane. Let’s say we want to draw an angle that’s 144° on our plane. We start on the right side of the x-axis, where three o’clock is on a clock. We rotate counterclockwise, which starts by moving up. We keep going past the 90° point (the top part of the y-axis) until we get to 144°. We draw a ray from the origin, which is the center of the plane, to that point. Now we have a ray that we call the terminal side. But we need to draw one more ray to make an angle. We have a choice at this point. Our second ray needs to be on the x-axis. If we draw it from the origin to the right side, we’ll have drawn an angle that measures 144°. If we draw it to the left, we’ll have drawn an angle that measures 36°. This second angle is the reference angle. It’s always the smaller of the two angles, will always be less than or equal to 90°, and it will always be positive. Here’s an animation that shows a reference angle for four different angles, each of which is in a different quadrant. Notice how the second ray is always on the x-axis.
The reference angle always has the same trig function values as the original angle. Notice the word values there. The sign may not be the same, but the value always will be. This is useful for common angles like 45° and 60° that we will encounter over and over again. Once we know their sine, cosine, and tangent values, we also know the values for any angle whose reference angle is also 45° or 60°. As for the sign, remember that Sine is positive in the 1st and 2nd quadrant and Cosine is positive in the 1st and 4th quadrant.
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When the terminal side is in the first quadrant (angles from 0° to 90°), our reference angle is the same as our given angle. This makes sense, since all the angles in the first quadrant are less than 90°. So, if our given angle is 33°, then its reference angle is also 33°.
When the terminal side is in the second quadrant (angles from 90° to 180°), our reference angle is 180° minus our given angle. So, if our given angle is 110°, then its reference angle is 180° – 110° = 70°.
When the terminal side is in the third quadrant (angles from 180° to 270°), our reference angle is our given angle minus 180°. So, if our given angle is 214°, then its reference angle is 214° – 180° = 34°.
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When the terminal side is in the fourth quadrant (angles from 270° to 360°), our reference angle is 360° minus our given angle. So, if our given angle is 332°, then its reference angle is 360° – 332° = 28°.
When an angle is greater than 360°, that means it has rotated all the way around the coordinate plane and kept on going. In order to find its reference angle, we first need to find its corresponding angle between 0° and 360°. This is easy to do. We just keep subtracting 360 from it until it’s below 360. For instance, if our angle is 544°, we would subtract 360° from it to get 184° (544° – 360° = 184°). Now we would notice that it’s in the third quadrant, so we’d subtract 180° from it to find that our reference angle is 4°.
When an angle is negative, we move the other direction to find our terminal side. This means we move clockwise instead of counterclockwise when drawing it. Or we can calculate it by simply adding it to 360°. For instance, if our given angle is –110°, then we would add it to 360° to find our positive angle of 250° (–110° + 360° = 250°). Now we would have to see that we’re in the third quadrant and apply that rule to find our reference angle (250° – 180° = 70°).
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This calculator can quickly find the reference angle, but in a pinch, remember that a quick sketch can help you remember the rules for calculating the reference angle in each quadrant.All articles published by are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by , including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to https:///openaccess.
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.
Editor’s Choice articles are based on recommendations by the scientific editors of journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.
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Research Center for Hyper-Connected Convergence Technology, School of ICT, Robotics and Mechanical Engineering, Institute of Information and Telecommunication Convergence (IITC), Hankyong National University, 327 Chungang-ro, Anseong 17579, Kyonggi-do, Republic of Korea
In this paper, we propose new three-dimensional (3D) visualization of objects at long distance under photon-starved conditions. In conventional three-dimensional image visualization techniques, the visual quality of three-dimensional images may be degraded because object images at long distances may have low resolution. Thus, in our proposed method, we utilize digital zooming, which can crop and interpolate the region of interest from the image to improve the visual quality of three-dimensional images at long distances. Under photon-starved conditions, three-dimensional images at long distances may not be visualized due to the lack of the number of photons. Photon counting integral imaging can be used to solve this problem, but objects at long distance may still have a small number of photons. In our method, a three-dimensional image can be reconstructed, since photon counting integral imaging with digital zooming is used. In addition, to estimate a more accurate three-dimensional image at long distance under photon-starved conditions, in this paper, multiple observation photon counting integral imaging (i.e., N observation photon counting integral imaging) is used. To show the feasibility of our proposed method, we implement the optical experiments and calculate performance metrics, such as peak sidelobe ratio. Therefore, our method can improve the visualization of three-dimensional objects at long distances under photon-starved conditions.
Three-dimensional (3D) visualization of objects at long distances on photon-starved conditions has been a great challenge in many applications, such as military, astronomy, and observing wild animals. In the military case, a defense or reconnaissance that searches enemies at long distances in the day or night is required. In astronomy, observing stars at billions of light years of distance is a critical problem. In addition, observing wild animals, which are nocturnal and have much wariness, is also needed.
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However, it is difficult to visualize the three-dimensional objects, which are located at long distances by conventional imaging methods, since lateral and longitudinal resolutions of the image at long distance may be reduced due to the limitation of optical devices and the image sensor. When a camera takes a picture, the object at long distance in the scene has less pixels than a close one. Therefore, lateral and longitudinal resolutions (i.e., three-dimensional resolution) of the image for objects at long distance are reduced. To visualize three-dimensional objects at long distance, integral imaging [1, 2, 3], which was first proposed by G. Lippmann, can be utilized. It uses two-dimensional (2D) images with different perspectives captured by lenslet array or camera array, where these images are referred to as elemental images. Integral imaging can provide full parallax and continuous viewing points of three-dimensional objects without any viewing glasses and coherent light sources [1, 2, 3, 4, 5, 6, 7, 8]. However, due to the limitation of three-dimensional resolution for three-dimensional objects at long distances, the visual quality of three-dimensional images at long distances may be degraded. In addition, this resolution problem may be critical under photon-starved conditions. Because an image sensor detects less photons, which have the information of an object at a long distance under photon-starved conditions, elemental images may not have the information of the object. That is, the visual quality of three-dimensional images may be more degraded under photon-starved conditions.
To visualize three-dimensional objects under photon-starved conditions, photon counting integral imaging [9, 10, 11] has been proposed. It can make a computational model of a photon detector by statistical
![PDF] Multi Level Tree Based Approach For Interactive Graph Visualization With Semantic Zoom - Zoomable Digital Art Reference Angle Charting](https://i0.wp.com/d3i71xaburhd42.cloudfront.net/6964edd776680ff48fcb5b25f3a83dc57228720e/4-Figure4-1.png?strip=all "PDF] Multi Level Tree Based Approach For Interactive Graph Visualization With Semantic Zoom")
This calculator can quickly find the reference angle, but in a pinch, remember that a quick sketch can help you remember the rules for calculating the reference angle in each quadrant.All articles published by are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by , including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to https:///openaccess.
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.
Editor’s Choice articles are based on recommendations by the scientific editors of journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.
Sony Fe Lenses: The Honest Guide For The A7/a9/a1 Series
Research Center for Hyper-Connected Convergence Technology, School of ICT, Robotics and Mechanical Engineering, Institute of Information and Telecommunication Convergence (IITC), Hankyong National University, 327 Chungang-ro, Anseong 17579, Kyonggi-do, Republic of Korea
In this paper, we propose new three-dimensional (3D) visualization of objects at long distance under photon-starved conditions. In conventional three-dimensional image visualization techniques, the visual quality of three-dimensional images may be degraded because object images at long distances may have low resolution. Thus, in our proposed method, we utilize digital zooming, which can crop and interpolate the region of interest from the image to improve the visual quality of three-dimensional images at long distances. Under photon-starved conditions, three-dimensional images at long distances may not be visualized due to the lack of the number of photons. Photon counting integral imaging can be used to solve this problem, but objects at long distance may still have a small number of photons. In our method, a three-dimensional image can be reconstructed, since photon counting integral imaging with digital zooming is used. In addition, to estimate a more accurate three-dimensional image at long distance under photon-starved conditions, in this paper, multiple observation photon counting integral imaging (i.e., N observation photon counting integral imaging) is used. To show the feasibility of our proposed method, we implement the optical experiments and calculate performance metrics, such as peak sidelobe ratio. Therefore, our method can improve the visualization of three-dimensional objects at long distances under photon-starved conditions.
Three-dimensional (3D) visualization of objects at long distances on photon-starved conditions has been a great challenge in many applications, such as military, astronomy, and observing wild animals. In the military case, a defense or reconnaissance that searches enemies at long distances in the day or night is required. In astronomy, observing stars at billions of light years of distance is a critical problem. In addition, observing wild animals, which are nocturnal and have much wariness, is also needed.
Pdf] Multi Level Tree Based Approach For Interactive Graph Visualization With Semantic Zoom
However, it is difficult to visualize the three-dimensional objects, which are located at long distances by conventional imaging methods, since lateral and longitudinal resolutions of the image at long distance may be reduced due to the limitation of optical devices and the image sensor. When a camera takes a picture, the object at long distance in the scene has less pixels than a close one. Therefore, lateral and longitudinal resolutions (i.e., three-dimensional resolution) of the image for objects at long distance are reduced. To visualize three-dimensional objects at long distance, integral imaging [1, 2, 3], which was first proposed by G. Lippmann, can be utilized. It uses two-dimensional (2D) images with different perspectives captured by lenslet array or camera array, where these images are referred to as elemental images. Integral imaging can provide full parallax and continuous viewing points of three-dimensional objects without any viewing glasses and coherent light sources [1, 2, 3, 4, 5, 6, 7, 8]. However, due to the limitation of three-dimensional resolution for three-dimensional objects at long distances, the visual quality of three-dimensional images at long distances may be degraded. In addition, this resolution problem may be critical under photon-starved conditions. Because an image sensor detects less photons, which have the information of an object at a long distance under photon-starved conditions, elemental images may not have the information of the object. That is, the visual quality of three-dimensional images may be more degraded under photon-starved conditions.
To visualize three-dimensional objects under photon-starved conditions, photon counting integral imaging [9, 10, 11] has been proposed. It can make a computational model of a photon detector by statistical
