CN112230837A - Method for changing viewing angle in three-dimensional dynamic geometric system - Google Patents
Method for changing viewing angle in three-dimensional dynamic geometric system Download PDFInfo
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- G06F3/0484—Interaction techniques based on graphical user interfaces [GUI] for the control of specific functions or operations, e.g. selecting or manipulating an object, an image or a displayed text element, setting a parameter value or selecting a range
- G06F3/04845—Interaction techniques based on graphical user interfaces [GUI] for the control of specific functions or operations, e.g. selecting or manipulating an object, an image or a displayed text element, setting a parameter value or selecting a range for image manipulation, e.g. dragging, rotation, expansion or change of colour
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- G06F3/0487—Interaction techniques based on graphical user interfaces [GUI] using specific features provided by the input device, e.g. functions controlled by the rotation of a mouse with dual sensing arrangements, or of the nature of the input device, e.g. tap gestures based on pressure sensed by a digitiser
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Abstract
S1 converts a screen coordinate system displayed by default when starting up into a standard coordinate system, S2 constructs a three-dimensional world coordinate system, an XOY surface of the world coordinate system is parallel to a screen, and the distance between the XOY surface of the world coordinate system and the screen is set; and the connection line of the world coordinate system origin and the standard coordinate system origin is vertical to the screen; the Z axis of the world coordinate system is vertical to the screen; and S3, constructing a three-dimensional system coordinate system in a world coordinate system, drawing a three-dimensional geometric figure in the three-dimensional system coordinate system, and changing the screen view angle of the three-dimensional geometric figure through changing the origin of the system coordinate system. The method realizes the on-screen dragging of the system coordinate system by dragging the origin of the system coordinate system, and the rotation of the visual angle and the scene depth zooming always take the three-dimensional system coordinate system origin as the center, thereby being convenient for showing the coordinate system transformation effect in the teaching.
Description
Technical Field
The invention belongs to the technical field of education, relates to geometric teaching software, and particularly relates to a method for changing a view angle in a three-dimensional dynamic geometric system.
Background
The dynamic geometry system is a teaching auxiliary system for basic mathematics education, and an interactive operating environment can be provided for teachers and students on the premise of keeping geometric and constraint algebraic relations in the system. The method is widely applied due to the characteristics of intuition, dynamics and the like.
Planar dynamic geometry systems are common today. Three-dimensional dynamic geometric systems are rare, technically more challenging, and more important for teaching assistance. In a three-dimensional dynamic system, the most fundamental problem is how to construct reasonable global coordinate system transformation so as to meet teaching habits and requirements and realize observation of a space geometry from different angles.
The prior art for changing the user's view angle to view a three-dimensional space from multiple angles is: and determining a coordinate system (generally a right-hand coordinate system) in the three-dimensional space to be kept unchanged, and changing parameters such as the position, the orientation, the normal direction and the like of the virtual camera through user interaction to realize the transformation of the user visual angle. The method specifically comprises the following steps:
1. fixing the camera viewpoint as the origin of a three-dimensional space coordinate system;
2. the camera rotates around a viewpoint to realize the rotation of a user visual angle to a space;
3. and adjusting the distance of the camera relative to the viewpoint to realize the zooming of the space by the user visual angle.
The viewpoint of the camera is fixed as the origin of the three-dimensional coordinate system, which is not beneficial to the translation operation of the three-dimensional coordinate system on the view angle. Namely, in the scheme, the origin of the three-dimensional coordinate system cannot move and is always kept at the center of the screen visually; and the transformation (rotation and zooming) of the viewing angle after the movement is still relative to the center of the screen visually, so that the requirements of teaching resource layout and operation cannot be met.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention discloses a method for changing the view angle in a three-dimensional dynamic geometric system.
The invention relates to a method for changing the view angle in a three-dimensional dynamic geometric system, which comprises the following steps:
s1, converting the screen coordinate system displayed by default when the computer is started into a standard coordinate system;
the screen coordinate system and the standard coordinate system are both two-dimensional coordinate systems positioned in a plane where the screen is positioned;
s2, constructing a three-dimensional world coordinate system, wherein an XOY surface of the world coordinate system is parallel to the screen, and the distance between the XOY surface of the world coordinate system and the screen is set; and the connection line of the world coordinate system origin and the standard coordinate system origin is vertical to the screen; the camera is placed in the world coordinate system and keeps the position unchanged;
s3, constructing a three-dimensional system coordinate system in a world coordinate system, drawing a three-dimensional geometric figure in the three-dimensional system coordinate system, and changing the screen view angle of the three-dimensional geometric figure through changing the origin of the system coordinate system.
Preferably, the origin of the standard coordinate system is the geometric center of the screen.
Preferably, in step S3, the transformation manner of any point a in the system coordinate system when the origin of the system coordinate system changes is as follows:
s31 calculates a photographing angle vector DR 2:
DR2=unproject（DR1,C）-OG
wherein C represents the coordinates of the camera in the standard coordinate system, OG represents the vector from the camera to the origin in the world coordinate system, DR1 is the vector from the point A to the origin in the standard coordinate system,
s32 shooting angle unit vector:
e = DR2/| DR2|, with | | denoting modulo.
S33；
Calculating the included angle between the OG and the e vector
cos(θ)=OG·e/|OG|·|e|
From this, the distance length dis of the camera position along the vector e to the plane of the world coordinate system XOY can be calculated as:
dis= cos(θ) ·|OG|
further calculating the intersection point of the vector e to the XOY plane of the world coordinate system
Inter=OG+dis·e
Inter is the coordinate (gx, gy, gz) of point A in the world coordinate system.
Preferably, the method further comprises rotating the system coordinate system, specifically;
moving the mouse, rotating the coordinate system according to the displacement of the mouse,
firstly, the vertical component of the mouse movement is converted into the y-axis rotation angle of the system coordinate system around the world coordinate system, and then the horizontal component of the mouse movement is converted into the z-axis rotation angle of the XOY plane of the system coordinate system around the system coordinate system.
Preferably, the method further comprises adjusting the scaling of the system coordinate system, specifically, rolling the mouse, and obtaining a scaling according to the rolling distance of the mouse; and changing the scaled value of the system coordinate system into the value of the current world coordinate system divided by the scaling, wherein the scaling center is the coordinate origin of the three-dimensional system coordinate system.
The system coordinate system is dragged on the screen by dragging the origin of the system coordinate system, the angle of rotation of the system coordinate system is modified by dragging the blank of the screen to change the angle of view, the system coordinate system is zoomed by the mouse roller to achieve the zoom of the scene depth, and the rotation of the angle of view and the zoom of the scene depth always take the three-dimensional origin of the system coordinate system as the center, so that the coordinate system transformation effect can be conveniently displayed in teaching.
Drawings
FIG. 1 is a schematic diagram of an embodiment of a screen coordinate system according to the present invention;
FIG. 2 is a diagram illustrating an embodiment of a standard coordinate system according to the present invention;
FIG. 3 is a schematic diagram of one embodiment of a world coordinate system according to the present invention;
FIG. 4 is a schematic diagram of an embodiment of a rotating system coordinate system according to the present invention;
FIG. 5 is a schematic diagram of another embodiment of a rotating system coordinate system according to the present invention;
FIG. 6 is a schematic diagram of another embodiment of a rotating system coordinate system according to the present invention.
Detailed Description
The following provides a more detailed description of the present invention.
The invention relates to a method for changing the view angle in a three-dimensional dynamic geometric system, which comprises the following steps:
s1, converting the screen coordinate system displayed by default when the computer is started into a standard coordinate system;
the screen coordinate system and the standard coordinate system are both two-dimensional coordinate systems positioned in a plane where the screen is positioned;
s2, constructing a three-dimensional world coordinate system, wherein an XOY surface of the world coordinate system is parallel to the screen, and the distance between the XOY surface of the world coordinate system and the screen is set; and the connection line of the world coordinate system origin and the standard coordinate system origin is vertical to the screen; the camera is placed in the world coordinate system and keeps the position unchanged;
s3, constructing a three-dimensional system coordinate system in a world coordinate system, drawing a three-dimensional geometric figure in the three-dimensional system coordinate system, and changing the screen view angle of the three-dimensional geometric figure through changing the origin of the system coordinate system.
The camera is fixed, and the position of the system coordinate system is updated in a zooming, translation and rotation mode. The transformation system coordinate system comprises the following elements:
a camera: the system is used for projecting the three-dimensional space to a two-dimensional screen, and the viewpoint is always the origin of a world coordinate system and is projected to the center of the screen; the camera position is fixed in the positive direction of the z axis of the world coordinate system, the negative direction of the x axis of the world coordinate system is located on the screen, and the camera position is located in the world coordinate system: (0, 0, camera z).
World coordinate system: a three-dimensional coordinate system actually used by the user;
user-defined graph: placing the three-dimensional geometric figure created by the user in a system coordinate system;
one specific operation process is as follows:
converting screen coordinates (Sx, Sy) in a screen coordinate system into standard coordinates (x, y) in a standard coordinate system, and assuming that the origin of the standard coordinate system is the geometric center of the screen, the width and the height of the screen are both defined as 2 in the standard coordinate system, namely the coordinate values of four vertexes of the screen in a world coordinate system are (1, 1), (1, -1), (-1, -1); the relationship between the screen coordinates and the standard coordinates is shown in fig. 1 and fig. 2: from the embodiment of fig. 1 and 2, the conversion formula of the screen coordinates in the screen coordinate system to the standard coordinates in the standard coordinate system is:
wherein width is the screen width and height is the screen height.
Constructing a three-dimensional world coordinate system, wherein an XOY surface of the world coordinate system is parallel to a screen, and the distance between the XOY surface of the world coordinate system and the screen is set; and the connection line of the world coordinate system origin and the standard coordinate system origin is vertical to the screen; the Z axis of the world coordinate system is vertical to the screen;
s3, constructing a three-dimensional system coordinate system in a world coordinate system, drawing a three-dimensional geometric figure in the three-dimensional system coordinate system, and changing the screen view angle of the three-dimensional geometric figure through changing the origin of the system coordinate system.
As shown in fig. 3, the observation depth of the camera actually varies with the variation of the viewing angle of the camera, the observation depth of the camera is deepest when the camera is observed perpendicular to the screen, and the depth that can be observed through the screen is limited by the physical size of the screen when the camera is observed obliquely.
The upper smaller rectangle in FIG. 3 represents the screen and a standard coordinate system located on the screen;
the larger rectangle located below represents the XOY plane of the world coordinate system, and the Z-axis in fig. 3 is the Z-axis of the world coordinate system. O1 and O2 denote the origin of the standard coordinate system and the world coordinate system, respectively, and C is the camera position.
And constructing a system coordinate system in the world coordinate system, wherein the graph to be observed is positioned in the system coordinate system, the original point of the system coordinate system is assumed as a point A, and the position of the system coordinate system on the screen is changed by changing the position of the point A in the world coordinate system.
One typical specific flow is:
and when the origin point changes, the system coordinate system changes because the camera always aims at the origin point of the system coordinate system, and the visual angle of the camera and the vector from the camera to the observation point change accordingly.
And (3) processing any point A in the system coordinate system as follows:
s31 calculates a photographing angle vector DR 2:
DR2=unproject（DR1,C）-OG
wherein C represents the coordinates of the camera in the standard coordinate system, OG represents the vector from the camera to the origin O2 in the world coordinate system, DR1 is the vector from the point A to the origin O1 in the standard coordinate system, and unprject (DR 1, C) represents the vector for projecting the DR1 vector in the standard coordinate system to the world coordinate system by taking the point C as the light source point;
s32 shooting angle unit vector:
e = DR2/| DR2|, which represents modulo.
S33；
Calculating the included angle between the OG and the e vector
cos(θ)=OG·e/|OG|·|e|
From this, the distance length dis of the camera position along the vector e to the plane of the world coordinate system XOY can be calculated as:
dis= cos(θ) ·|OG|
further calculating the intersection point of the vector e to the XOY plane of the world coordinate system
Inter=OG+dis·e
Inter is the new coordinate (gx, gy, gz) of the point A in the world coordinate system.
And traversing each point needing to be converted in the system coordinate system, such as each point on the three-dimensional geometric figure in the system coordinate system, so as to obtain the converted figure displayed on the screen under the change of the system coordinate system and the camera view angle.
By fixing the position of the camera and setting the origin of the camera to be always aligned with the coordinate system of the system, when the coordinate system of the system is converted, a user can more clearly see the change of the system coordinate system, which leads to the change of the visual angle of the graph inside the coordinate system, and the graph can always occupy the central position of the screen, thereby avoiding the graph from being moved to the corner of the screen or out of the screen due to the rotation of the coordinate system in other processing modes.
In the invention, the system coordinate system can be rotated,
a schematic diagram of the coordinate system of the rotating system is shown in fig. 4. In fig. 4, WX, WY, and WZ are X, Y, Z axes of the world coordinate system, respectively. SX, SY and SZ are X, Y, Z axes before the system coordinate system rotates. SX1, SY1, SZ1 are X, Y, Z axes after the system coordinate system is rotated.
In fig. 4, the Z axis of the system coordinate system before rotation coincides with the Z axis of the world coordinate system, and after rotation by a certain angle using the Y axis of the world coordinate system as a rotation axis, the Y axis of the system coordinate system after rotation coincides with the Y axis of the world coordinate system.
The specific operation process is as follows: pressing and moving the mouse, wherein the components of the mouse moving on the screen in the vertical direction and the horizontal direction are dy and dx respectively;
the XOY plane of the world coordinate system is always parallel to the screen and the Z-axis is perpendicular to the screen. The system coordinate system typically does not overlap with the world coordinate system.
The system coordinate system is rotated by dy radians around the y-axis of the world coordinate system. A denominator may be set to scale the rotation angle, for example, the rotation angle is set to dy/100, that is, the final rotation angle is determined by dividing the mouse movement component by 100, and the specific unit conversion may be determined by the mouse sensitivity, the screen size, and other factors.
The system coordinate system is rotated by an angle of dy/100 degrees with the y-axis of the world coordinate system as a rotation axis.
The system coordinate system is SX1, SY1 and SZ1 in FIG. 4.
The system coordinate system is rotated by dx radians around its z-axis,
that is, the system coordinate system is rotated by dx/100 radians with the z-axis of the system coordinate system as the axis of rotation.
Because the whole rotation process is the rotation angle of the changed system coordinate system relative to the world coordinate system, and the center position of the system coordinate system is always fixed, the rotation center is always determined to be the origin of the system coordinate system during the rotation process and the rotation system.
Since the Z axis of the world coordinate system is perpendicular to the screen and the Y axis is displayed in a vertical state on the screen, the above rotation mode of rotating with the Y axis of the world coordinate system as the axis is also more suitable for teaching habits and student observation, and the Z axis of the system coordinate system is always kept in a vertical direction on the screen during rotation transformation, as shown in fig. 5 and 6.
The operation of scaling the system coordinate system may be specifically: rolling the mouse, and obtaining a scaling radio according to the rolling distance of the mouse; the scaled value of the modified system coordinate system is the value of the current world coordinate system divided by radio. For example, when the original length of a certain line segment in the world coordinate system is 10, and the zoom ratio radio =2 is obtained by mouse scrolling, after zooming, the unit length of the system coordinate system is reduced to half relative to the world coordinate system, the three-dimensional display graph in the system coordinate system is also reduced in an equal proportion, and the zoom center during zooming is always kept as the coordinate origin of the system coordinate system.
The system coordinate system is dragged on the screen by dragging the origin of the system coordinate system, the angle of rotation of the system coordinate system is modified by dragging the blank of the screen to change the angle of view, the system coordinate system is zoomed by the mouse roller to achieve the zoom of the scene depth, and the rotation of the angle of view and the zoom of the scene depth always take the three-dimensional origin of the system coordinate system as the center, so that the coordinate system transformation effect can be conveniently displayed in teaching.
The foregoing is directed to preferred embodiments of the present invention, wherein the preferred embodiments are not obviously contradictory or subject to any particular embodiment, and any combination of the preferred embodiments may be combined in any overlapping manner, and the specific parameters in the embodiments and examples are only for the purpose of clearly illustrating the inventor's invention verification process and are not intended to limit the scope of the invention, which is defined by the claims and the equivalent structural changes made by the description and drawings of the present invention are also intended to be included in the scope of the present invention.
Claims (5)
1. A method for transforming a view angle in a three-dimensional dynamic geometric system is characterized by comprising the following steps:
s1, converting the screen coordinate system displayed by default when the computer is started into a standard coordinate system;
the screen coordinate system and the standard coordinate system are both two-dimensional coordinate systems positioned in a plane where the screen is positioned;
s2, constructing a three-dimensional world coordinate system, wherein an XOY surface of the world coordinate system is parallel to the screen, and the distance between the XOY surface of the world coordinate system and the screen is set; and the connection line of the world coordinate system origin and the standard coordinate system origin is vertical to the screen; the camera is placed in the world coordinate system and keeps the position unchanged;
s3, constructing a three-dimensional system coordinate system in a world coordinate system, drawing a three-dimensional geometric figure in the three-dimensional system coordinate system, and changing the screen view angle of the three-dimensional geometric figure through changing the origin of the system coordinate system.
2. The method as claimed in claim 1, wherein the origin of the standard coordinate system is the geometric center of the screen.
3. The method for transforming a viewing angle in a three-dimensional dynamic geometrical system according to claim 1, wherein the transformation manner of any point a in the system coordinate system when the origin of the system coordinate system changes in step S3 is as follows:
s31 calculates a photographing angle vector DR 2:
DR2=unproject（DR1,C）-OG
wherein C represents the coordinates of the camera in the standard coordinate system, OG represents the vector from the camera to the origin in the world coordinate system, DR1 is the vector from the point A to the origin in the standard coordinate system,
s32 shooting angle unit vector:
e = DR2/| DR2|, with | | denoting modulo;
S33；
calculating the included angle between the OG and the e vector
cos(θ)=OG·e/|OG|·|e|
From this, the distance length dis of the camera position along the vector e to the plane of the world coordinate system XOY can be calculated as:
dis= cos(θ) ·|OG|
further calculating the intersection point of the vector e to the XOY plane of the world coordinate system
Inter=OG+dis·e
Inter is the coordinate (gx, gy, gz) of point A in the world coordinate system.
4. The method for transforming a view angle in a three-dimensional dynamic geometrical system according to claim 1, wherein the method further comprises rotating the system coordinate system, in particular;
moving the mouse, rotating the coordinate system according to the displacement of the mouse,
firstly, the vertical component of the mouse movement is converted into the y-axis rotation angle of the system coordinate system around the world coordinate system, and then the horizontal component of the mouse movement is converted into the z-axis rotation angle of the XOY plane of the system coordinate system around the system coordinate system.
5. The method of claim 1, further comprising scaling the system coordinate system, specifically by scrolling a mouse, to obtain a scaling according to a distance of the mouse scrolling; and changing the scaled value of the system coordinate system into the value of the current world coordinate system divided by the scaling, wherein the scaling center is the coordinate origin of the three-dimensional system coordinate system.
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