JP5645463B2 - Method for converting a three-dimensional shape into a three-dimensional shape - Google Patents
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Description
本発明は、従来の直線、円、球などの曲線、曲面、3次元形状から、この変換の元となる数を用いて、3次元の新たな曲線、曲面、3次元形状、3次元形状を得ることができる。また、時間の項も追加することにより、3次元形状の動画を3次元形状の動画に変換することができる。すなわち、簡単な3次元形状または3次元動画をこの変換Tで変換することにより、複雑で滑らかな3次元形状または3次元動画を得ることができる。
また、得られた複雑な曲線、曲面、3次元形状、3次元動画は元の形状と変換Tにより、簡単に表現でき、コンピュータ上の記憶領域を少なくすることを目的とする。コンピュータ上で、複雑な3次元曲線を描く場合、多くの点が必要で、その点間をスプライン曲線のような式で補間することになる。また、3次元曲面を描く場合も、同様に多くの点が必要でその構成する点を補間することになる。複雑な3次元曲線、3次元曲面を正確に滑らかに描く場合、より多くの点が必要になる。この変換前の形状は通常、簡単な形状のため、必要な点は少なく、少ない点と変換Tを少ない記憶領域でコンピュータ上に記憶するだけで、実際に必要な時に簡単に計算できるので、すぐに複雑な3次元曲線、3次元曲面が画面上に描画できる。変換式も簡単なため、高速に変換し、描画することができる。当然ながら、3次元CAD上で、いろいろな角度から3次元の曲線、3次元曲面を見ることができる。3次元形状は幾つかの3次元曲面で構成されているので、同様である。
According to the present invention, a new three-dimensional curve, curved surface, three-dimensional shape, or three-dimensional shape is converted from a conventional curve, curved surface, or three-dimensional shape such as a circle, a sphere, or the like using the number that is the basis of this conversion. Can be obtained. Further, by adding a term of time, a three-dimensional moving image can be converted into a three-dimensional moving image. That is, by converting a simple three-dimensional shape or a three-dimensional moving image by this conversion T, a complicated and smooth three-dimensional shape or a three-dimensional moving image can be obtained.
Further, the object is to reduce the storage area on the computer by easily expressing the obtained complex curve, curved surface, three-dimensional shape, and three-dimensional moving image by the original shape and conversion T. When drawing a complicated three-dimensional curve on a computer, many points are required, and the point is interpolated with an expression like a spline curve. Similarly, when drawing a three-dimensional curved surface, a large number of points are necessary and the constituent points are interpolated. More points are required to draw complex 3D curves and 3D curved surfaces accurately and smoothly. Since the shape before conversion is usually a simple shape, there are few points required, and since it is easy to calculate when it is actually needed, it can be calculated easily when it is actually needed by storing a small number of points and the conversion T on a computer with a small storage area. 3D curved and 3D curved surfaces can be drawn on the screen. Since the conversion formula is also simple, it can be converted and drawn at high speed. Of course, you can see 3D curves and 3D curved surfaces from various angles on 3D CAD. The 3D shape is the same because it is composed of several 3D curved surfaces.
従来はジューフスコフスキー変換というものがあり、複素数zを用いて、
ω=z+bの2乗/zという変換で、飛行機の翼の形状が得られるような変換があった。しかしながら、これは2次元形状しか得られず、実際の翼の形状のように3次元の翼の形状は得られなかった。そこで、あらたに複素数の複素数なるものを定義し(仮想的複素数)、3次元から3次元の変換ができるようになった。
Conventionally, there is a so-called Jufskovsky transformation, using a complex number z,
There was a conversion in which the shape of the wing of an airplane was obtained by the conversion of ω = z + b squared / z. However, only a two-dimensional shape can be obtained, and a three-dimensional blade shape like an actual blade shape cannot be obtained. Therefore, a new complex number is defined (virtual complex number), and three-dimensional to three-dimensional conversion can be performed.
ジューフスコフスキー変換は複素数を用いた2次元から2次元の変換であるが、4次元から4次元の変換をするために拡張した。
そこで、iの2乗 =−1なる複素数iを用いてtの2乗=iを定義し、
T=a+bi+ct (a、b、cは実数)なる数で表される3次元の直交座標は
x=a、y=b、z=cで表される。この数を用いて、TからTへの変換を考えると3次元から4次元への変換が得られ、4次元目のitの項を無視すると、変換後の形状を用いて、新たな3次元の形状を得る。
The Jufskovsky transformation is a two-dimensional to two-dimensional transformation using complex numbers, but has been extended to perform a four-dimensional to four-dimensional transformation.
So i squared Define the square of t = i using the complex number i = 1
A three-dimensional orthogonal coordinate represented by a number T = a + bi + ct (where a, b, and c are real numbers) is represented by x = a, y = b, and z = c. When conversion from T to T is considered using this number, conversion from 3D to 4D is obtained, and if the fourth term it term is ignored, a new 3D is obtained using the converted shape. Get the shape.
また、iの2乗 =−1なる複素数iを用いてmの2乗=−iを定義し、
T=a+bi+cm(a、b、cは実数)なる数で表される3次元の直交座標は
x=a
、y=b 、z=cで表される。この数を用いて、TからTへの変換を考えると3次元から4次元への変換が得られ、4次元目のimの項を無視すると、変換後の形状を用いて、新たな3次元の形状を得る。ここで、t、mを仮想的複素数と呼ぶことにする。
I squared Define the square of m = −i using the complex number i == − 1,
The three-dimensional orthogonal coordinates represented by the number T = a + bi + cm (a, b, and c are real numbers) are
x = a
, Y = b and z = c. When conversion from T to T is considered using this number, conversion from 3D to 4D is obtained, and if the term of im in the 4th dimension is ignored, a new 3D is obtained using the converted shape. Get the shape. Here, t and m are referred to as virtual complex numbers.
T=a+bi+ct+dit(a、b、c、dは実数)なる数で表される数は、3次元の直交座標を
x=a
、y=b 、z=cで表わし、時間の項はdで表す。この数を用いて、TからTへの変換を考えると4次元から4次元への変換が得られ、時間d=0から始まった3次元形状が時間項itを含む変換になり、変換後は新たな3次元の時間項を持つ動画を得る。
The number represented by the number T = a + bi + ct + dit (where a, b, c, and d are real numbers) is a three-dimensional orthogonal coordinate.
x = a
, Y = b and z = c, and the term of time is represented by d. Considering the conversion from T to T using this number, the conversion from 4D to 4D is obtained, and the 3D shape starting from time d = 0 becomes the conversion including the time term it. Get a video with a new 3D time term.
また、iの2乗 =−1なる複素数iを用いてmの2乗=−iを定義し、
T=a+bi+cm+ dim(a、b、c、dは実数)なる数で表される数は、3次元の直交座標をx=a、y=b
、z=cで表わし、時間の項はdで表す。この数を用いて、TからTへの変換を考えると4次元から4次元への変換が得られ、時間d=0から始まった3次元形状が時間項itを含む変換になり、変換後は新たな3次元の時間項を持つ動画を得る。
I squared Define the square of m = −i using the complex number i == − 1,
The number represented by the number T = a + bi + cm + dim (where a, b, c, and d are real numbers) is a three-dimensional orthogonal coordinate x = a, y = b
, Z = c, and the term of time is represented by d. Considering the conversion from T to T using this number, the conversion from 4D to 4D is obtained, and the 3D shape starting from time d = 0 becomes the conversion including the time term it. Get a video with a new 3D time term.
特許出願平10−37753でも変換を行っているが、これは3次元の物体をモニタなどの2次元の表示への変換の問題であって、本特許は3次元から3次元への3次元形状から新しい3次元形状、3次元の形状を得るための変換であって、問題としている目的、方法も異なる。 Patent application Hei 10-37753 also performs conversion, but this is a problem of converting a three-dimensional object into a two-dimensional display such as a monitor. This patent describes a three-dimensional shape from three dimensions to three dimensions. Conversion to obtain a new three-dimensional shape from the three-dimensional shape, and the object and method in question are also different.
複素数で表される変換は2次元から2次元であるが、これを複素数の複素数なる仮想的複素数を定義し、3次元動画から3次元動画への変換を可能にした。
ジューフコフスキー変換も複素数から複素数の変換の一種であるが、簡単な円等の形状から複雑な翼の断面形状を得ていた。しかしながら、2次元の形状しか得られなかった。本発明の変換Tにおいては、簡単な3次元の形状が複雑な3次元の形状に変換されるので、簡単な3次元の形状をCAD上で表現すれば、複雑な3次元形状が得られ、変換前の形状、変換のTに対して工夫すれば、ジューフコフスキー変換が2次元で必要な形状が得られたのと同様に、必要な3次元形状が得られる。通常、コンピュータ上で複雑な3次元形状は多くの点、曲面から構成され、記憶領域が多く必要であるが、変換前の3次元形状と変換Tを記憶するだけでよく、必要な時に再度計算して、表示すればいいので、保存の時に記憶領域が少なくて済む。変換式も簡単なため、高速に変換し、描画することができる。
The conversion represented by complex numbers is from two dimensions to two dimensions, but this is defined as a virtual complex number, which is a complex number of complex numbers, to enable conversion from a three-dimensional moving image to a three-dimensional moving image.
The Jufkovsky transformation is a kind of complex-to-complex transformation, but it has obtained complex wing cross-sectional shapes from simple shapes such as circles. However, only a two-dimensional shape was obtained. In the conversion T of the present invention, since a simple three-dimensional shape is converted into a complicated three-dimensional shape, if a simple three-dimensional shape is expressed on CAD, a complicated three-dimensional shape can be obtained. If the shape before conversion and T of the conversion are devised, the required three-dimensional shape can be obtained in the same manner as the shape required for the two-dimensional Jufkovsky transformation. Normally, a complicated three-dimensional shape on a computer is composed of many points and curved surfaces and requires a large storage area, but it is only necessary to store the three-dimensional shape before conversion and the conversion T, and recalculate when necessary. Since it only has to be displayed, the storage area can be reduced at the time of saving. Since the conversion formula is also simple, it can be converted and drawn at high speed.
iの2乗=−1なる複素数iを用いてtの2乗=iを定義し、
T=a+bi+ct(a、b、cは実数、時間it=0)なる数を考える3次元の直交座標はx=a
、y=b 、z=cで表される。この数を用いて、TからTへの変換を考えると3次元から4次元への変換が得られ、4次元目のitの項を無視すると、変換後の形状を用いて、新たな3次元の形状を得る。
Define the square of t = i using the complex number i of the square of i = −1,
A three-dimensional orthogonal coordinate considering a number T = a + bi + ct (a, b, c are real numbers, time it = 0) is x = a
, Y = b and z = c. When conversion from T to T is considered using this number, conversion from 3D to 4D is obtained, and if the fourth term it term is ignored, a new 3D is obtained using the converted shape. Get the shape.
また、iの2乗 =−1なる複素数iを用いてmの2乗=−iを定義し、
T=a+bi+cm(a、b、cは実数、時間im=0)なる数を考える3次元の直交座標はx=a 、y=b 、z=cで表される。この数を用いて、TからTへの変換を考えると3次元から4次元への変換が得られ、4次元目のimの項を無視すると、変換後の形状を用いて、新たな3次元の形状を得る。
I squared Define the square of m = −i using the complex number i = −1,
A three-dimensional orthogonal coordinate considering a number T = a + bi + cm (a, b, c are real numbers, time im = 0) is represented by x = a, y = b, z = c. When conversion from T to T is considered using this number, conversion from 3D to 4D is obtained, and if the term of im in the 4th dimension is ignored, a new 3D is obtained using the converted shape. Get the shape.
T=a+bi+ct+dit(a、b、c、dは実数)なる数で表される数は、3次元の直交座標を
x=a
、y=b 、z=cで表わし、時間の項はdで表す。この数を用いて、TからTへの変換を考えると4次元から4次元への変換が得られ、時間d=0から始まった3次元形状が時間項iTを含む変換になり、変換後は新たな3次元の時間項を持つ動画を得る。
また、iの2乗 =−1なる複素数i
を用いてm の2乗=−iを定義し、
T=a+bi+cm+dim(a、b、c、dは実数)なる数で表される数は、3次元の直交座標をx=a
、y=b 、z=cで表わし、時間の項はdで表す。この数を用いて、TからTへの変換を考えると4次元から4次元への変換が得られ、時間d=0から始まった3次元形状が時間項itを含む変換になり、変換後は新たな3次元の時間項を持つ動画を得る。
The number represented by the number T = a + bi + ct + dit (where a, b, c, and d are real numbers) is a three-dimensional orthogonal coordinate.
x = a
, Y = b and z = c, and the term of time is represented by d. Considering the conversion from T to T using this number, the conversion from 4D to 4D is obtained, and the 3D shape starting from time d = 0 becomes the conversion including the time term iT. Get a video with a new 3D time term.
I squared = Complex number i
Define m squared = -i using
The number represented by the number T = a + bi + cm + dim (where a, b, c, and d are real numbers) is a three-dimensional orthogonal coordinate x = a
, Y = b and z = c, and the term of time is represented by d. Considering the conversion from T to T using this number, the conversion from 4D to 4D is obtained, and the 3D shape starting from time d = 0 becomes the conversion including the time term it. Get a video with a new 3D time term.
本発明は、従来の直線、円、球などの曲線、曲面、3次元形状から、この変換の元となる数を用いて、3次元の新たな曲線、曲面、3次元形状、3次元形状を得ることができる。また、時間の項も追加することにより、3次元形状の動画を3次元形状の動画に変換することができる。すなわち、簡単な3次元形状または3次元動画をこの変換Tで変換することにより、複雑で滑らかな3次元形状または3次元動画を得ることができる。
また、得られた複雑な曲線、曲面、3次元形状、3次元動画は元の形状と変換Tにより、簡単に表現でき、コンピュータ上の記憶領域を少なくすることを目的とする。コンピュータ上で、複雑な3次元曲線を描く場合、多くの点が必要で、その点間をスプライン曲線のような式で補間することになる。また、3次元曲面を描く場合も、同様に多くの点が必要でその構成する点を補間することになる。複雑な3次元曲線、3次元曲面を正確に滑らかに描く場合、より多くの点が必要になる。この変換前の形状は通常、簡単な形状のため、必要な点は少なく、少ない点と変換Tを少ない記憶領域でコンピュータ上に記憶するだけで、実際に必要な時に簡単に計算できるので、すぐに複雑な3次元曲線、3次元曲面が画面上に描画できる。変換式も簡単なため、高速に変換し、描画することができる。当然ながら、3次元CAD上で、いろいろな角度から3次元の曲線、3次元曲面を見ることができる。3次元形状は幾つかの3次元曲面で構成されているので、同様である。
According to the present invention, a new three-dimensional curve, curved surface, three-dimensional shape, or three-dimensional shape is converted from a conventional curve, curved surface, or three-dimensional shape such as a circle, a sphere, or the like using the number that is the basis of this conversion. Can be obtained. Further, by adding a term of time, a three-dimensional moving image can be converted into a three-dimensional moving image. That is, by converting a simple three-dimensional shape or a three-dimensional moving image by this conversion T, a complicated and smooth three-dimensional shape or a three-dimensional moving image can be obtained.
Further, the object is to reduce the storage area on the computer by easily expressing the obtained complex curve, curved surface, three-dimensional shape, and three-dimensional moving image by the original shape and conversion T. When drawing a complicated three-dimensional curve on a computer, many points are required, and the point is interpolated with an expression like a spline curve. Similarly, when drawing a three-dimensional curved surface, a large number of points are necessary and the constituent points are interpolated. More points are required to draw complex 3D curves and 3D curved surfaces accurately and smoothly. Since the shape before conversion is usually a simple shape, there are few points required, and since it is easy to calculate when it is actually needed, it can be calculated easily when it is actually needed by storing a small number of points and the conversion T on a computer with a small storage area. 3D curved and 3D curved surfaces can be drawn on the screen. Since the conversion formula is also simple, it can be converted and drawn at high speed. Of course, you can see 3D curves and 3D curved surfaces from various angles on 3D CAD. The 3D shape is the same because it is composed of several 3D curved surfaces.
T=a+bi+ct(時間項it=0)で、a、b、cをそれぞれx、y、zの座標とする。
ω=Tの2乗
の変換を計算すると
ω=(a+bi+ct)( a+bi+ct)=aの2乗+abi+act +abi
+bの2乗×iの2乗
+bcit +act+bcit+cの2乗×tの2乗
=(aの2乗−bの2乗)+(2ab+cの2乗)i+2act+2bcit
よって、x座表x:a→(aの2乗−bの2乗)
y座表y:b→(2ab+cの2乗)
z座標z:c→2ac
時間項 it=2bcという項が出てきて、実際は3次元から4次元になっているが、空間から空間からの変換を考える場合は、itの項は無視する。
T = a + bi + ct (time term it = 0), and a, b, and c are coordinates of x, y, and z, respectively.
When calculating the transformation of the square of ω = T, ω = (a + bi + ct) (a + bi + ct) = the square of a + abi + act + abi
+ B squared x i squared
+ Bcit + act + bcit + c squared × t squared = (a squared−b squared) + (2ab + c squared) i + 2act + 2 bcit
Therefore, x coordinate table x: a → (a squared−b squared)
y coordinate y: b → (2ab + c squared )
z coordinate z: c → 2ac
A term of time term it = 2 bc comes out and is actually 3D to 4D. However, when considering conversion from space to space, the term of it is ignored.
これで、Z=0 xy座標で中心x=y=0で半径1の円を変換すると2周する惰円になる(図2)。
Z=1平面上のxy座標が中心x=y=0で半径1の円を変換すると複雑な曲線になる(図3)。角度を変えて見た図が図4から図6になる。座標系も図中に示す。変換前の曲線、変換Tが決まれば、このような3次元の曲線は3次元CADで容易に描画できる。変換Tが重要な項目になる。
中心x=y=z=0の半径1の球(図7)を変換すると富士山のような形状になる。
変換前の球の等高線が図6で、変換後の球の等高線を図7に示す。図8の球の等高線を別の角度から見た図が図9から図12である。座標系も図中に示す。このような3次元形状とその変換後の形状も3次元CADで容易に描画できる。変換Tが重要な項目になる。
すなわち、変換前の形状は簡単な形状なので、コンピュータ上でも簡単に表現でき、少ない記憶領域ですむ。変換後の形状は複雑な形状なので、通常、点を多く取り、微小な線分の集まりで曲線を表現し、曲面の場合、その微小な線分の集まりで一つの微小な曲面を定義し、その微小な曲面のあつまりで、全体の曲面を表現する。図示した(図2〜12)曲線も3次元CADにこの変換Tが組み込まれていないので、同様の複雑な処置をしている。このように通常では変換後の図形をコンピュータ上で記憶するには多くの記憶領域を必要とする。
また、図示した(図2〜12)ように2次元の用紙には2次元の図形しか表現できない。このように、通常のCADソフトで画面を見るようにxyz座標系において、ある位置から見た図形をある面に投影すると2次元形状が得られる。また、ある方向からある面に投影すれば2次元形状が得られる。
Thus, when a circle with a radius of 1 at the center x = y = 0 with Z = 0 xy coordinates is converted, a circle with two rounds is formed (FIG. 2).
When a circle with a xy coordinate of the center x = y = 0 on the Z = 1 plane and a radius of 1 is converted, a complex curve is obtained (FIG. 3). Figures viewed from different angles are shown in FIGS. The coordinate system is also shown in the figure. If the curve before conversion and the conversion T are determined, such a three-dimensional curve can be easily drawn by three-dimensional CAD. The conversion T becomes an important item.
When a sphere with radius 1 at the center x = y = z = 0 (FIG. 7) is transformed, it becomes a shape like Mt. Fuji.
The contour lines of the sphere before conversion are shown in FIG. 6, and the contour lines of the sphere after conversion are shown in FIG. FIGS. 9 to 12 show the contour lines of the sphere of FIG. 8 as seen from different angles. The coordinate system is also shown in the figure. Such a three-dimensional shape and the converted shape can be easily drawn with a three-dimensional CAD. The conversion T becomes an important item.
In other words, since the shape before conversion is simple, it can be easily expressed on a computer and requires a small storage area. Since the shape after conversion is a complicated shape, usually a large number of points are taken, a curve is expressed by a collection of minute line segments, and in the case of a curved surface, one minute curved surface is defined by the collection of minute line segments, The entire curved surface is expressed by the small curved surface. The illustrated curve (FIGS. 2 to 12) does not incorporate this transformation T in the three-dimensional CAD, and thus performs the same complicated procedure. As described above, usually, a large number of storage areas are required to store the converted figure on the computer.
Further, as shown (FIGS. 2 to 12), only a two-dimensional figure can be expressed on a two-dimensional sheet. In this way, a two-dimensional shape can be obtained by projecting a figure viewed from a certain position onto a certain surface in the xyz coordinate system so that the screen is viewed with normal CAD software. A two-dimensional shape can be obtained by projecting from a certain direction onto a certain surface.
Tの変換は幾つかあって、
ジューフスコフスキー変換を拡張したものとして、
T=T+bの2乗/T も考えられる。
There are several conversions of T
As an extension of the Jufskovsky transformation,
T = T + b squared / T can also be considered.
また、it、im を時間の項として、T=a+bi+ct+dit またはT=a+bi+cm+dimと定義すると3次元の動画を表現でき、a、b、c、dはそれぞれ、x、y、z座標、時間と考えればよい。この場合、足し算、引き算、掛け算は可能で、d=0の時のみ割り算が可能である。
3次元から3次元の変換が必要な場合は、変換前はd=0 で表せ、変換後it、imの項がでてきてもその項は時間として考え、使わないで表現すれば、3次元から3次元への変換が可能である。
Also, if it and im are defined as terms of time and T = a + bi + ct + dit or T = a + bi + cm + dim, a three-dimensional moving image can be expressed. Good. In this case, addition, subtraction, and multiplication are possible, and division is possible only when d = 0.
When conversion from 3D to 3D is required, it can be expressed by d = 0 before conversion, and even if it and im terms appear after conversion, those terms can be considered as time and expressed without using 3D. To three-dimensional conversion is possible.
また、d=0の時、Tの割り算が可能である。
このTの変換を考える時、数Tに対して、aTや(aT+bT)の2乗
などの変換など足し算、引き算、掛け算のいずれの変換を用いてもよい。但し、数Tに対して割り算は1回のみで1/Tを含む変換のみである。すなわち、割り算には制限があるが、四則演算が可能で、適宜な式を用いてよい。いずれの変換においても、適宜、時間を表す項、it、im
の項を無視すれば、3次元T0=a+bi+ct またはT1=a+bi+cmで3次元形状を表し、Tの何らかの変換を与えれば、新たな3次元形状が得られる。
また、このような種種の四則演算を行ったTの変換も通常の最大値や最小値や平均を求める函数と同様に簡単に計算できるので、変換Tにおいてもコンピュータ上のプログラムで簡単に表現でき、記憶領域を多く必要としない。
また、eの(iπ)乗=−1(e:自然対数の底、i:複素数)なる従来の関係があるが、この場合
(tの2乗×mの2乗)=i×(−i)=−iの2乗=1より、
t×m=±1
よって、
eの(tの2乗×mの2乗)= e (t、m:上記で定義した仮想的複素数)
eの(t×m×i×π)=eの(±1×i×π)=−1
なる関係が成り立つ。これを前田拓也の式と呼ぶことにする。
Further, when d = 0, T can be divided.
When considering the conversion of T, any conversion of addition, subtraction, or multiplication, such as conversion such as the square of aT or (aT + bT), may be used for the number T. However, the division with respect to the number T is performed only once, and only conversion including 1 / T is performed. That is, although division is limited, four arithmetic operations are possible, and an appropriate formula may be used. In any conversion, terms representing time, it, im, as appropriate
If this term is ignored, a three-dimensional shape is expressed by three-dimensional T0 = a + bi + ct or T1 = a + bi + cm, and if any transformation of T is given, a new three-dimensional shape is obtained.
In addition, since the transformation of T with these various arithmetic operations can be easily calculated in the same way as a normal maximum value, minimum value, or average function, the transformation T can be easily expressed by a computer program. Does not require a lot of storage space.
In addition, there is a conventional relationship of e raised to the power of (iπ) = − 1 (e: base of natural logarithm, i: complex number).
(square of t × square of m) = i × (−i) = − square of −i = 1
t × m = ± 1
Therefore,
e ((square of t × square of m)) = e (t, m: virtual complex number defined above)
(t × m × i × π) of e = (± 1 × i × π) of e = −1
The relationship becomes true. This is called the Maeda Takuya's ceremony.
3次元の動画から動画への変換の場合、T=a+bi+ct+dit またはT=a+bi+cm+dimで表せるが、割り算をする場合は、初期条件として、d=0になる。
すなわち、最初は時間0である3次元形状になる。その後は、3次元形状はTの変換によって、時間の項が現れる。
In the case of conversion from a three-dimensional moving image to a moving image, it can be expressed by T = a + bi + ct + dit or T = a + bi + cm + dim, but when dividing, d = 0 is set as an initial condition.
In other words, the three-dimensional shape is initially time 0. After that, the term of time appears in the three-dimensional shape by the conversion of T.
この変換により、簡単な形状が複雑な形状になる。
すなわち、簡単な3次元形状または3次元動画をこの変換Tで変換することにより、複雑で滑らかな3次元形状または3次元動画をえることができる。
また、変換前の簡単な3次元形状または3次元動画と変換Tから、複雑な3次元形状または3次元動画が得られる。通常、複雑な3次元形状、3次元動画をコンピュータに記録するためには多くの記憶領域が必要であるが、変換前の簡単な3次元形状、3次元動画と変換Tのみの記憶のみなら、記憶領域を削減できる。
具体的には、得られた複雑な曲線、曲面、3次元形状、3次元動画は元の形状と変換Tにより、簡単に表現でき、コンピュータ上の記憶領域を少なくすることを目的とする。コンピュータ上で、複雑な3次元曲線を描く場合、多くの点が必要で、その点間をスプライン曲線のような式で補間することになる。また、3次元曲面を描く場合も、同様に多くの点が必要でその構成する点を補間することになる。複雑な3次元曲線、3次元曲面を正確に滑らかに描く場合、より多くの点が必要になる。この変換前の形状は通常、簡単な形状のため、必要な点は少なく、少ない点と変換Tを少ない記憶領域でコンピュータ上に記憶するだけで、実際に必要な時に簡単に計算できるので、すぐに複雑な3次元曲線、3次元曲面が画面上に描画できる。変換式も簡単なため、高速に変換し、描画することができる。当然ながら、3次元CAD上で、いろいろな角度から3次元の曲線、3次元曲面を見ることができる。3次元形状は幾つかの3次元曲面で構成されているので、同様である。
By this conversion, a simple shape becomes a complicated shape.
That is, by converting a simple three-dimensional shape or a three-dimensional moving image by this conversion T, a complicated and smooth three-dimensional shape or a three-dimensional moving image can be obtained.
Further, a complicated three-dimensional shape or three-dimensional moving image can be obtained from the simple three-dimensional shape or three-dimensional moving image before conversion and the conversion T. Usually, a large amount of storage space is required to record a complicated 3D shape and 3D video on a computer, but if only a simple 3D shape before conversion, 3D video and conversion T are stored, Storage area can be reduced.
Specifically, an object is to reduce the storage area on the computer by easily expressing the obtained complex curve, curved surface, three-dimensional shape, and three-dimensional moving image by the original shape and transformation T. When drawing a complicated three-dimensional curve on a computer, many points are required, and the point is interpolated with an expression like a spline curve. Similarly, when drawing a three-dimensional curved surface, a large number of points are necessary and the constituent points are interpolated. More points are required to draw complex 3D curves and 3D curved surfaces accurately and smoothly. Since the shape before conversion is usually a simple shape, there are few points required, and since it is easy to calculate when it is actually needed, it can be calculated easily when it is actually needed by storing a small number of points and the conversion T on a computer with a small storage area. 3D curved and 3D curved surfaces can be drawn on the screen. Since the conversion formula is also simple, it can be converted and drawn at high speed. Of course, you can see 3D curves and 3D curved surfaces from various angles on 3D CAD. The 3D shape is the same because it is composed of several 3D curved surfaces.
本発明は、従来の直線、円、球などの曲線、曲面、3次元形状から、この変換の元となる数を用いて、3次元の新たな曲線、曲面、3次元形状、3次元形状を得ることができる。また、時間の項も追加することにより、3次元形状の動画を3次元形状の動画に変換することができる。すなわち、簡単な3次元形状または3次元動画をこの変換Tで変換することにより、複雑で滑らかな3次元形状または3次元動画を得ることができる。
また、得られた複雑な曲線、曲面、3次元形状、3次元動画は元の形状と変換Tにより、簡単に表現でき、コンピュータ上の記憶領域を少なくすることを目的とする。コンピュータ上で、複雑な3次元曲線を描く場合、多くの点が必要で、その点間をスプライン曲線のような式で補間することになる。また、3次元曲面を描く場合も、同様に多くの点が必要でその構成する点を補間することになる。複雑な3次元曲線、3次元曲面を正確に滑らかに描く場合、より多くの点が必要になる。この変換前の形状は通常、簡単な形状のため、必要な点は少なく、少ない点と変換Tを少ない記憶領域でコンピュータ上に記憶するだけで、実際に必要な時に簡単に計算できるので、すぐに複雑な3次元曲線、3次元曲面が画面上に描画できる。変換式も簡単なため、高速に変換し、描画することができる。当然ながら、3次元CAD上で、いろいろな角度から3次元の曲線、3次元曲面を見ることができる。3次元形状は幾つかの3次元曲面で構成されているので、同様である。
According to the present invention, a new three-dimensional curve, curved surface, three-dimensional shape, or three-dimensional shape is converted from a conventional curve, curved surface, or three-dimensional shape such as a circle, a sphere, or the like using the number that is the basis of this conversion. Can be obtained. Further, by adding a term of time, a three-dimensional moving image can be converted into a three-dimensional moving image. That is, by converting a simple three-dimensional shape or a three-dimensional moving image by this conversion T, a complicated and smooth three-dimensional shape or a three-dimensional moving image can be obtained.
Further, the object is to reduce the storage area on the computer by easily expressing the obtained complex curve, curved surface, three-dimensional shape, and three-dimensional moving image by the original shape and conversion T. When drawing a complicated three-dimensional curve on a computer, many points are required, and the point is interpolated with an expression like a spline curve. Similarly, when drawing a three-dimensional curved surface, a large number of points are necessary and the constituent points are interpolated. More points are required to draw complex 3D curves and 3D curved surfaces accurately and smoothly. Since the shape before conversion is usually a simple shape, there are few points required, and since it is easy to calculate when it is actually needed, it can be calculated easily when it is actually needed by storing a small number of points and the conversion T on a computer with a small storage area. 3D curved and 3D curved surfaces can be drawn on the screen. Since the conversion formula is also simple, it can be converted and drawn at high speed. Of course, you can see 3D curves and 3D curved surfaces from various angles on 3D CAD. The 3D shape is the same because it is composed of several 3D curved surfaces.
Claims (3)
T=a+bi+ct(a、b、cは実数)なる数Tを考えると、3次元の直交座標は
x=a 、 y=b 、 z=c で表される。この数Tに対して関数(Tの2乗 )または、関数(T+pの2乗/T:pは実数)による変換により数T1が得られる。
この数Tと上記関数により得られたT1により、3次元から3次元への変換が得られ、変換後の形状を用いて、3次元の形状を得るCADにおけるグラフィックス方法i squared Define the square of t = i using the imaginary number i = 1
Considering the number T as T = a + bi + ct (a, b, and c are real numbers), the three-dimensional orthogonal coordinates are represented by x = a, y = b, and z = c. The number T1 is obtained by converting the number T by a function (T squared ) or a function (T + p squared / T: p is a real number).
A graphics method in CAD that obtains a three-dimensional shape using the number T and T1 obtained from the above function and obtains a three-dimensional transformation from the transformed shape.
T=a+bi+cm (a、b、cは実数)なる数Tを考えると、3次元の直交座標は
x=a 、 y=b 、 z=c で表される。この数Tに対して関数(Tの2乗 )または、関数(T+pの2乗/T:pは実数)による変換により数T2が得られる。
この数Tと上記関数により得られたT2により、3次元から3次元への変換が得られ、変換後の形状を用いて、3次元の形状を得るCADにおけるグラフィックス方法i squared Define m squared = -i using imaginary number i = 1
Considering the number T as T = a + bi + cm (where a, b, and c are real numbers), the three-dimensional Cartesian coordinates are represented by x = a, y = b, and z = c. The number T2 is obtained by conversion of the number T by a function (T squared ) or a function (T + p squared / T: p is a real number).
A graphics method in CAD that obtains a three-dimensional shape using the number T and T2 obtained from the above function, and obtains a three-dimensional transformation from the transformed shape.
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