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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