Question

You are given the three-dimensional coordinates of a point \(P 1\left(x_{1}, y_{1}, z_{1}\right)\) and a point \(P 2\left(x_{2}, y_{2}, z_{2}\right)\). You are also given the coordinates of the location point of a viewer \(\left(x_{v}, y_{v}, z_{1}\right)\). You may assume that \(\mathrm{P} 1\) and \(\mathrm{P} 2\) are located on the same side of the viewer. Describe informally (you do not need to write out an algorithm) exactly how to determine if, from the point of view of the viewer, it is possible to see both points P1 and \(\mathrm{P} 2\), or if one of these points is obstructed and not visible. In the latter case, describe how you can determine which is the occluded point.

Short answer

Determine sight lines to both points and check for overlap; the further one is the occluded point.

Step-by-step solution

Understanding the Viewer Position

First, recognize that the viewer is located at the point \((x_v, y_v, z_1)\). Importantly, note that the viewer shares the same z-coordinate as \(P1\) and \(P2\), meaning they are all in the same plane along the z-axis. This will be crucial for assessing visibility since any obstruction would occur within this three-dimensional space.

Forming Lines of Sight

Determine the lines of sight from the viewer to each point. For \(P1\), the line of sight can be described by the vector directed from \((x_v, y_v, z_1)\) to \(P1(x_1, y_1, z_1)\). For \(P2\), follow a similar process creating a vector from the viewer's point to \(P2(x_2, y_2, z_2)\). The coordinates of the vector for \(P1\) are \((x_1 - x_v, y_1 - y_v, 0)\) and for \(P2\), \((x_2 - x_v, y_2 - y_v, z_2 - z_1)\).

Determine Potential Blockage

Conceptualize the situation geometrically: if neither point is in the direct path of the line to the other, both should be visible. Imagine projecting lines from the viewer's location to \(P1\) and \(P2\). Concretely, assess if the line segments overlap or intersect in a way that either \(P1\) is on the line between the viewer and \(P2\) or vice versa. If \(P1\) lies directly on the path from the viewer to \(P2\) or \(P2\) lies directly on the path from the viewer to \(P1\), check which point lies further along that line from the viewer, to identify which point blocks the other. You can verify this by comparing the parameter values (t) assuming parametric line equations for blocking lines.

Check Visibility by Distance

If Step 3 suggests the possibility of occlusion, calculate the Euclidean distance from the viewer to each point: for \(P1\), \(\sqrt{(x_1-x_v)^2 + (y_1-y_v)^2 + (z_1-z_1)^2}\) and for \(P2\), \(\sqrt{(x_2-x_v)^2 + (y_2-y_v)^2 + (z_2-z_1)^2}\). If the distances are distinct, the point with the greater distance from the viewer is the one potentially occluded.

Key concepts

Visibility Analysis

Visibility analysis involves understanding which objects in a 3D space can be seen from a particular point or perspective. In our scenario, we are trying to determine the visibility of points \(P1\) and \(P2\) from a viewer's location. To do this, you consider the line of sight, which is the imaginary line connecting the viewer to each point.
By evaluating these lines and their potential intersections, you can assess whether one point is obstructing the view of another. This is crucial in applications such as computer graphics and architecture to ensure seamless imagery and to understand how different elements of a scene may become visible or hidden as the viewer moves.
When conducting visibility analysis, the spatial relationships between objects are critical. It's about understanding how objects align, overlap, or intersect in 3D space as projected onto a 2D plane from the viewer's perspective.

Viewer Perspective

The viewer's perspective is central to determining what is and isn't visible in a 3D setting. This concept involves considering where the viewer is located and how this position influences the scene's visibility.
In our example, the viewer is positioned at \((x_v, y_v, z_1)\), sharing a z-coordinate plane with both \(P1\) and \(P2\). This shared plane simplifies the perspective, as any changes in visibility will result from the positions' relative distances and alignments, not from elevation differences.
Understanding the viewer's perspective involves imagining yourself at that point and visualizing the lines extending from you to other objects. It's similar to how a camera's position and angle affect what it can capture in a frame. In the viewer's perspective, the closer an object is along the line of sight, the more prominently it features, which can either lead to it blocking another object or being blocked itself.

Occlusion Detection

Occlusion detection refers to identifying when one object blocks the view of another. This concept is vital for realistic rendering in virtual environments.
In our problem, occlusion occurs if one point, say \(P1\), is directly in line with and between the viewer and another point, \(P2\). When this happens, \(P1\) occludes (or blocks) \(P2\) from the viewer's line of sight.
Detecting occlusion involves analyzing the geometric alignments, which can be visualized by projecting lines from the viewer to each point and checking for intersections. When implementing this, you can use parametric equations for the lines as a mathematical way to check if the point along one line coincides with another.

• If the parametric values indicate overlap and \(P1\) or \(P2\) lies within the line segment connecting the viewer to the other, an occlusion occurs.
• Calculate the distances from the viewer to get a sense of which point is nearer, as the further one would likely be the occluded point.

Coordinate System

A coordinate system, in this context, is the framework that describes the positions of points in three-dimensional space using three numbers: \((x, y, z)\).
These coordinates allow you to navigate and visualize space, making it crucial for analyzing visibility and perspective. The coordinate system in our exercise helps define the vector lines from the viewer to points \(P1\) and \(P2\), providing the scalar values needed to assess occlusion.
In many visualization tasks, the choice of coordinate system can impact your ability to analyze the spatial relationships effectively. It could be Cartesian, spherical, or another form, but here the Cartesian system is used because it straightforwardly represents distance and direction in uniform dimensions.

• This system enables calculations that determine lines of sight by simply subtracting component-wise to get vectors.
• The coordinate system also makes it possible to compute distances using the Euclidean formula, which is crucial for deciding potential occlusion based on how far each point is from the viewer.