CN113281902B - Ray and optical free-form surface intersection point position iterative solution method based on secant method - Google Patents
Ray and optical free-form surface intersection point position iterative solution method based on secant method Download PDFInfo
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Abstract
A secant method-based iterative solution method for intersection positions of light rays and an optical free-form surface belongs to the field of optical design. The method comprises the following steps: step one, determining two initial iteration points on an optical free-form surface; step two, determining a point on the light ray which is closest to the secant space distance; step three, determining a next iteration point on the optical free-form surface; step four, judging an iteration exit condition; step five, updating the two initial iteration points; and step six, continuously completing the iterative solving process of the intersection point position of the light and the optical free-form surface according to the step three to the step five. The invention has the advantages of simple calculation process, small calculation amount, high solving efficiency, high solving precision, convenient program compiling, maintenance and implementation and the like.
Description
Technical Field
The invention belongs to the technical field of optical design, and particularly relates to a secant method-based intersection point position iteration solving method for light and an optical free-form surface.
Background
Optical simulation software is widely applied in the field of optical design. At present, most of optical simulation software used in optical design is mainly researched and developed by foreign countries, and the optical design of China also mostly depends on the foreign optical simulation software. With the continuous development of optical design technology, the research and development of a kind of home-made optical simulation software is imminent.
The key point of the development of the optical simulation software lies in a ray tracing algorithm, which belongs to a bottom layer module of the optical simulation software and determines the performance of the whole optical simulation software. The ray tracing process involves two crucial steps: solving the intersection point position of the light ray (straight line) and the optical free-form surface and calculating the direction of the refracted/reflected light ray. The optical free-form surface is widely applied in the optical field at present, but the mathematical description form is complex, the intersection point position of the inclined light (straight line) and the optical free-form surface cannot be directly solved in an analytic mode, and iterative solution is needed. And the iterative solution algorithm determines the efficiency and accuracy of solving the intersection point position of the light (straight line) and the optical free-form surface.
The optical free-form surface has a plurality of complex forms such as XY polynomial, zernike polynomial, and Q-Type polynomial. However, in the conventional iterative process for solving the intersection point of the light ray and the optical free-form surface, a tangent plane of a certain point on the optical free-form surface needs to be calculated first, and then the next iteration point on the optical free-form surface is determined by calculating the intersection point of the light ray and the tangent plane. Because different expression forms of the optical free-form surface are different and even complex, the calculation process of the next iteration point on the optical free-form surface becomes complex, the calculation amount is large, and the solving efficiency and the solving precision are reduced. Meanwhile, when the intersection point of the light and the optical free-form surface is solved for different optical free-form surfaces, corresponding different iteration programs need to be written, which is not beneficial to the maintenance and implementation of the iteration programs.
Disclosure of Invention
The invention provides a ray and optical free-form surface intersection point position iterative solution method based on a secant method, and aims to solve the problems of complex calculation process, large calculation amount, low efficiency and precision and inconvenience in programming and maintenance of the existing ray and optical free-form surface intersection point position solution method.
The technical scheme adopted by the invention for solving the technical problem is as follows:
the invention relates to a secant method-based ray and optical free-form surface intersection point position iterative solution method, which comprises the following steps:
step one, determining two initial iteration points on an optical free-form surface;
step two, determining a point on the light ray which is closest to the secant space distance;
step three, determining a next iteration point on the optical free-form surface;
step four, judging an iteration exit condition;
step five, updating the two initial iteration points;
and step six, continuously completing the iterative solving process of the intersection point position of the light and the optical free-form surface according to the step three to the step five.
Further, the specific operation process of the step one is as follows:
the first method for determining the initial iteration point comprises the following steps:
(1) determining a tangent plane of a substantially spherical portion of the optical free-form surface at a vertex thereof;
(2) determining an intersection point M of the light and the tangent plane;
(3) the intersection point A of the straight line passing through the intersection point M along the optical axis direction and the optical free-form surface is a first initial iteration point of the optical free-form surface;
(4) the coordinates (x) of the first initial iteration point Aa,ya) Coordinate (x) with the intersection point Mm,ym) Similarly, the coordinate z of the initial iteration point A is calculated according to the curved vector height equation (1)a；
The second method for determining the initial iteration point comprises the following steps:
(1) determining an intersection point N of the light ray and a basic spherical part of the optical free-form surface;
(2) the intersection point B of the straight line passing through the intersection point N along the optical axis direction and the optical free-form surface is the second initial iteration point of the optical free-form surface 2;
(3) the coordinates (x) of the second initial iteration point Bb,yb) Coordinate (x) with the intersection point Nn,yn) Similarly, the coordinate z of the iteration point B is calculated according to the curved rise equation (1)b；
Wherein z (x, y) represents the profile height at coordinates (x, y); the rise of the face shape is divided into two parts, the first part
Denotes a substantially spherical portion of the optical free-form surface, c denotes a substantially spherical portion curvature of the optical free-form surface; the second portion F (x, y) is an optical free-form surface portion in a broad sense, and represents the amount of deviation of the actual optical free-form surface portion from the substantially spherical portion.
Further, the specific operation process of the step two is as follows:
(1) the equations of the light rays and the secant lines are converted into symmetrical equations, and direction vectors n1 ═ a1, b1 and c1, n2 ═ a2, b2 and c2 are respectively obtained;
(2) the common vector of the two direction vectors n1 and n2 is obtained by cross-multiplying the two direction vectors: N-N1 × N2;
(3) and solving a point on the light ray which is closest to the space distance of the secant by utilizing the condition and combining an equation simultaneous equation of the light ray and the secant, wherein the point is set as Q.
Further, the step two is replaced by the following steps:
taking the point on the cutting line closest to the light ray space as a point Q:
the equation of the light ray and the secant itself is a symmetrical equation, and a direction vector n1 is obtained (a1, b1, c1), and n2 is obtained (a2, b2, c 2);
the common vector of the two direction vectors n1 and n2 is obtained by cross-multiplying the two direction vectors: N-N1 × N2;
and thirdly, solving a point on the secant line closest to the light ray space by utilizing the condition and combining with an equation simultaneous equation of the light ray and the secant line, wherein only one straight line in the straight lines connecting one point on the secant line and one point on the light ray is parallel to the common vertical quantity N, namely N1 multiplied by N2, and the distance between the point on the secant line and the point on the light ray is the shortest distance between the secant line and the light ray.
Further, the step two is replaced by the following steps:
one of two initial iteration points on the optical free-form surface is utilized to form a plane with the light, and the intersection point of the projection straight line of the secant on the plane and the light is taken as a point Q.
Further, the specific operation process of the step three is as follows:
the crossing point Q is a straight line parallel to the optical axis, and the intersection point C of the straight line and the optical free-form surface is the next iteration point on the optical free-form surface; coordinates (x) of iteration point Cc,yc) Coordinate (x) with point Qq,yq) Similarly, the iteration point is calculated according to the curved rise equation (1)C coordinate zc。
Further, the specific operation process of the step four is as follows:
and calculating the distance between the point Q and the iteration point C, comparing the distance with a certain threshold, and if the distance is smaller than the threshold, exiting the iteration, wherein the iteration point C is the final iteration result.
Further, the specific operation process of the step five is as follows:
and if the distance between the point Q and the iteration point C is larger than the threshold value, updating the two iteration points, namely marking the iteration point B as an iteration point A and marking the iteration point C as an iteration point B.
The invention has the beneficial effects that:
the intersection point position iterative solution method of the ray and the optical free-form surface based on the secant method is used for tracing the ray so as to determine the propagation path of the ray in the optical system. The invention can carry out fast iterative solution on the intersection point position of the light (straight line) and the optical free-form surface in a manner of easy programming realization.
Compared with the prior art, the invention has the following technical effects:
1. the calculation process is simple and the calculation amount is small. In the process of iterative solution of intersection point positions of the light rays and the optical free-form surface, the method does not need to solve the tangent plane at a certain intermediate iteration point on the optical free-form surface, saves a complex calculation process of the tangent plane of the optical free-form surface, greatly reduces the calculation amount, and has the advantages of simple calculation process and small calculation amount.
2. The solving efficiency is high. According to the method, the single iteration convergence rate is not high, however, the calculation amount required by the single iteration is small, so that the method can carry out more iterations within the same time, and has high iteration solving efficiency.
3. The solving precision is high. Because the invention can realize more iterations in the same time, the precision of iterative solution can be greatly improved.
4. And the writing, maintenance and implementation of the program are convenient. The intersection points for solving different optical free-form surfaces can all adopt the same programming form, so that the program has higher efficiency and is easy to write and maintain.
Drawings
Fig. 1 is a process for determining two initial iteration points on an optical free-form surface according to the present invention.
FIG. 2 is a process of determining a third iteration point from the first two initial iteration points in the present invention.
FIG. 3 is a flowchart of an iterative solution method for intersection positions of rays and an optical free-form surface based on a secant method.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Although the intersection point between the light ray and the optical free-form surface cannot be calculated by using an analytical formula, a point on the optical free-form surface close enough to the real intersection point can be obtained in a certain iterative mode. Based on this, the method for iteratively solving the intersection point position of the light ray and the optical free-form surface based on the secant method, as shown in fig. 3, specifically comprises the following steps:
1. determination of two initial iteration points on optical free-form surface 2
The rise of the optical free-form surface in the optical axis (z-axis) direction can be generally expressed as:
the above equation (1) is a curved vector height equation. Wherein z (x, y) represents the face vector height at coordinate (x, y); the profile rise can be divided into two parts, the first part on the right side of the equation
Presentation opticsA basic spherical portion of the free-form surface (c represents a curvature of the basic spherical portion of the optical free-form surface), which determines first-order parameters of the optical free-form surface such as a focal length and a magnification; the second part F (x, y) on the right side of the equation may be referred to as an optical free-form surface part in a broad sense, and represents the deviation amount of the actual optical free-form surface part from the substantially spherical surface part.
The ray tracing process is actually a process of transferring one optical free-form surface to another optical free-form surface. Therefore, in the present invention, two optical free-form surfaces are set, that is, light is emitted from the optical free-form surface 1 to the optical free-form surface 2.
As shown in fig. 1, the determination method of the first initial iteration point is as follows:
(1) a tangent plane of the substantially spherical portion 3 of the optical free-form surface 2 at its apex is determined.
(2) The intersection M of the ray and the tangent plane is determined (the ray equation is known, and the intersection M with a plane is easily obtained).
(3) An intersection point a of a straight line passing through the intersection point M in the optical axis (z axis) direction and the optical free-form surface 2 is a first initial iteration point of the optical free-form surface 2.
(4) The coordinates (x) of the first initial iteration point Aa,ya) Coordinate (x) with the intersection point Mm,ym) Similarly, the coordinate z of the initial iteration point A can be obtained according to the curved surface rise equation (1)a。
As shown in fig. 1, the determination method of the second initial iteration point is as follows:
(1) an intersection point N of the light ray with the basic spherical portion 3 of the optical free-form surface 2 is determined (both the light ray and the spherical equation are known, and the intersection point N is easily obtained).
(2) An intersection point B of a straight line passing through the intersection point N along the optical axis (z axis) direction and the optical free-form surface 2 is a second initial iteration point of the optical free-form surface 2.
(3) The coordinates (x) of the second initial iteration point Bb,yb) Coordinate (x) with the intersection point Nn,yn) Similarly, the coordinate z of the iteration point B can be obtained according to the curved surface rise equation (1)b。
2. Determination of the point on the ray that is closest in spatial distance to the secant (secant refers to the straight line connecting two adjacent iteration points on the optical free-form surface 2)
After determining the two initial iteration points, the next iteration point needs to be found. At present, the existing methods are: firstly, calculating a tangent plane at an initial iteration point on the optical free-form surface, and then determining a next iteration point on the optical free-form surface by calculating an intersection point of a ray and the tangent plane. However, the determination of the tangent plane requires gradient calculation, the calculation process is complicated, and the calculation formula depends on the specific optical free-form surface description form.
The tangential direction passing through a certain point on the optical free-form surface can be approximately replaced by the direction of a certain secant line passing through the point (a connecting line between the point and another point near the point on the optical free-form surface), and the conclusion can be popularized to a three-dimensional curved surface from a two-dimensional curve.
The secant passing through two points on the optical free-form surface and the light generally do not have an intersection point in a three-dimensional space, but the secant and the light have the nearest distance in the space and are unique.
The invention provides that the point on the light ray which is closest to the space distance of the secant is used for replacing the intersection point of the light ray and the tangent plane at a certain iteration point on the optical free-form surface 2, wherein the light ray is not intersected with the secant nor is parallel generally. The specific process for determining this point is as follows:
(1) the equations of the light ray and the secant themselves are symmetric expressions, and direction vectors n1 ═ a1, b1, and c1 ═ n2 ═ a2, b2, and c2 are obtained.
(2) The common vector of the two direction vectors n1 and n2 is obtained by cross-multiplying the two direction vectors: N-N1 × N2.
(3) Because one and only one straight line in the straight lines connecting one point on the light ray and one point on the secant (the light ray and the secant do not intersect and are not parallel) is parallel to the common vertical quantity N which is N1 multiplied by N2, the distance between the point on the light ray and the point on the secant is the shortest distance between the light ray and the secant, the point on the light ray which is closest to the space distance of the secant is solved by utilizing the condition and combining the equations of the light ray and the secant and the equations of the light ray and the secant, and the point is set as Q.
Based on the secant thought, the secant connecting two adjacent iteration points on the optical free-form surface 2 replaces a tangent plane of a certain point on the optical free-form surface 2 to determine the next iteration point. One particular problem is that there is generally no intersection between the secant and the ray in three-dimensional space. The invention proposes to use the point on the ray closest to the secant space as a generalized "intersection point" (i.e., point Q), and in fact, there are other methods of determining point Q, including:
(1) taking the point on the cutting line closest to the light ray space as a point Q; referring to the method, the specific process is as follows:
the equations of the light ray and the secant themselves are symmetric equations, and direction vectors n1 are (a1, b1, c1), and n2 are (a2, b2, c 2).
The common vector of the two direction vectors n1 and n2 is obtained by cross-multiplying the two direction vectors: N-N1 × N2.
And thirdly, solving a point on the secant, which is closest to the light ray space distance, by using the condition and combining equations of the light ray and the secant itself, and setting the point as Q, wherein only one straight line in the straight lines connecting one point on the secant and one point on the light ray (assuming that the light ray and the secant do not intersect and are not parallel) is parallel to the common vertical quantity N of the straight lines, namely N1 multiplied by N2, and the distance between the point on the secant and the point on the light ray is the shortest distance between the secant and the light ray.
(2) One of two initial iteration points on the optical free-form surface 2 and the light ray form a plane, and the intersection point of the projection straight line of the secant on the plane and the light ray is taken as a point Q.
3. Determination of the next iteration point on the optical free-form surface 2
The process of determining the third iteration point C from the two initial iteration points A, B is shown in fig. 2. Where the point Q represents a point on the light ray that is closest in spatial distance to the secant (a straight line connecting the adjacent two initial iteration points A, B on the optical free-form surface 2). Since FIG. 2 is a two-dimensional diagram, where the display point Q is the intersection of a ray and a secant, in actual three-dimensional space, the ray and the secant generally do not intersect nor are they parallel.
On the determined lightAfter the point Q closest to the secant space distance, a straight line (x and y coordinates are kept unchanged) parallel to the optical axis (z axis) is made through the point Q, and the intersection point C with the optical free-form surface 2 is the next iteration point on the optical free-form surface 2 (the z coordinate is easily obtained by using a surface vector height equation). Coordinates (x) of iteration point Cc,yc) Coordinate (x) with point Qq,yq) Similarly, the coordinate z of the iteration point C can be obtained according to the curved surface rise equation (1)c。
4. Determination of an iteration exit condition
And calculating the distance between the point Q and the iteration point C, comparing the distance with a certain threshold, and if the distance (the length of the line segment QC) is less than the threshold, exiting the iteration, wherein the iteration point C is the final iteration result. The threshold is a parameter for measuring the iterative solution accuracy of the intersection position of the light ray and the optical free-form surface, for example, the threshold can be set to 1e-10, and the threshold can be set according to actual needs.
5. Updating of two initial iteration points
And if the distance between the point Q and the iteration point C is larger than the threshold value, updating the two iteration points, namely marking the iteration point B as an iteration point A and marking the iteration point C as an iteration point B.
6. And (5) continuously completing the iterative solving process of the intersection point position of the light ray and the optical free-form surface according to the step (3), the step (4) and the step (5).
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.
Claims (3)
1. The method for iteratively solving the intersection point position of the light and the optical free-form surface based on the secant method is characterized by comprising the following steps of:
step one, determining two initial iteration points on an optical free-form surface;
the specific operation process of the first step is as follows:
the first method for determining the initial iteration point comprises the following steps:
(1) determining a tangent plane of a substantially spherical portion of the optical free-form surface at a vertex thereof;
(2) determining an intersection point M of the light and the tangent plane;
(3) the intersection point A of the straight line passing through the intersection point M along the optical axis direction and the optical free-form surface is a first initial iteration point of the optical free-form surface;
(4) the coordinates (x) of the first initial iteration point Aa,ya) Coordinate (x) with the intersection point Mm,ym) Similarly, the coordinate z of the initial iteration point A is calculated according to the curved vector height equation (1)a；
The second method for determining the initial iteration point comprises the following steps:
(1) determining an intersection point N of the light ray and a basic spherical part of the optical free-form surface;
(2) the intersection point B of the straight line passing through the intersection point N along the optical axis direction and the optical free-form surface is the second initial iteration point of the optical free-form surface 2;
(3) the coordinates (x) of the second initial iteration point Bb,yb) Coordinate (x) with the intersection point Nn,yn) Similarly, the coordinate z of the iteration point B is calculated according to the curved rise equation (1)b；
Wherein z (x, y) represents the profile height at coordinates (x, y); the rise of the face shape is divided into two parts, the first part
Denotes a substantially spherical portion of the optical free-form surface, c denotes a substantially spherical portion curvature of the optical free-form surface; the second portion F (x, y) is an optical free-form surface portion in a broad sense, and represents a deviation amount of an actual optical free-form surface portion from a substantially spherical surface portion;
step two, determining a point on the light ray which is closest to the secant space distance;
the specific operation process of the second step is as follows:
(1) the equations of the light rays and the secant lines are converted into symmetrical equations, and direction vectors n1 ═ a1, b1 and c1, n2 ═ a2, b2 and c2 are respectively obtained;
(2) the common vector of the two direction vectors n1 and n2 is obtained by cross-multiplying the two direction vectors: N-N1 × N2;
(3) one straight line and only one straight line in the straight lines connecting one point on the light ray and one point on the secant are parallel to the common vertical quantity N which is N1 multiplied by N2, the distance between the point on the light ray and the point on the secant is the shortest distance between the light ray and the secant, the point on the light ray which is closest to the space distance of the secant is solved by utilizing the condition and combining the simultaneous equations of the light ray and the secant and the equation of the light ray and the secant, and the point is set as Q;
step three, determining a next iteration point on the optical free-form surface;
the specific operation process of the third step is as follows:
the crossing point Q is a straight line parallel to the optical axis, and the intersection point C of the straight line and the optical free-form surface is the next iteration point on the optical free-form surface; coordinates (x) of iteration point Cc,yc) Coordinate (x) with point Qq,yq) Similarly, the coordinate z of the iteration point C is calculated according to the curved rise equation (1)c；
Step four, judging an iteration exit condition;
the specific operation process of the step four is as follows:
calculating the distance between the point Q and the iteration point C, comparing the distance with a certain threshold, and if the distance is smaller than the threshold, exiting the iteration, wherein the iteration point C is the final iteration result;
step five, updating the two initial iteration points;
the concrete operation process of the step five is as follows:
if the distance between the point Q and the iteration point C is larger than the threshold value, updating two iteration points, namely recording the iteration point B as an iteration point A and recording the iteration point C as an iteration point B;
and step six, continuously completing the iterative solving process of the intersection point position of the light and the optical free-form surface according to the step three to the step five.
2. The method for iteratively solving intersection positions of rays and the optical free-form surface based on the secant method according to claim 1, wherein the step two is replaced by the following steps:
taking the point on the cutting line closest to the light ray space as a point Q:
the equation of the light ray and the secant itself is a symmetrical equation, and a direction vector n1 is obtained (a1, b1, c1), and n2 is obtained (a2, b2, c 2);
the common vector of the two direction vectors n1 and n2 is obtained by cross-multiplying the two direction vectors: N-N1 × N2;
and thirdly, solving a point on the secant line closest to the light ray space by utilizing the condition and combining with an equation simultaneous equation of the light ray and the secant line, wherein only one straight line in the straight lines connecting one point on the secant line and one point on the light ray is parallel to the common vertical quantity N, namely N1 multiplied by N2, and the distance between the point on the secant line and the point on the light ray is the shortest distance between the secant line and the light ray.
3. The method for iteratively solving intersection positions of rays and the optical free-form surface based on the secant method according to claim 1, wherein the step two is replaced by the following steps:
one of two initial iteration points on the optical free-form surface is utilized to form a plane with the light, and the intersection point of the projection straight line of the secant on the plane and the light is taken as a point Q.
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