XYZ 3D Scatterplot

Includes: 3D scatter plot (X,Y,Z), optional group coloring, optional point labels, rotation/zoom + plane projections, optional wireframe 3D objects (sphere / ellipsoid), axis reversal, manual axis limits, marker-symbol selection, and animated GIF export.
Purpose: Visualize three-dimensional relationships with interactive view controls, optional wireframe geometry overlays, and optional animation output.

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Overview

The XYZ 3D Scatter Plot tool creates an interactive 3D-looking scatter plot in Excel by projecting 3D points \((x,y,z)\) onto a 2D chart canvas, then drawing a bounding “cage”, tick marks, optional gridlines, optional plane projections, and optional Z-drop lines.

Important

This is not Excel’s native 3D chart type. BESH Stat NG builds an XY scatter chart and draws a 3D effect using a deterministic 3D→2D projection.

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When to use

Use XYZ 3D Scatter Plot when you want to:

• visually assess relationships among three continuous variables;
• see clusters or separation across groups (via Group ID);
• add point identifiers (via Datapoint labels);
• enhance depth perception with plane projections and Z-drop lines.

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Example dataset

This example uses the first three variables from the scatter plot matrix example dataset:

• X = Expect
• Y = Entertain
• Z = Comm

Dataset: 012scatterplotmatrix.csv Dataset for animated GIF: 011xyzscatterplot_gif.csv

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User interface

The dialog is organized into three tabs:

• Input
• 3D Objects
• Axis Options

Input tab

Data inputs

• X, Y, Z (required): numeric ranges of equal length.
• Group ID (optional): a categorical variable (text or numeric) used to color points and create a legend.
• Datapoint labels (optional): labels shown next to points when Show Data Point Labels is enabled.
• Animated Gif Inputs (optional): a 6-column numeric table defining one animation frame per row.

Missing values: rows with missing values in any selected input range (X, Y, Z, and optional Group ID / labels) are removed listwise.

View settings

• Rotation X axis [degree] (default 120): rotates the view about the X axis.
• Rotation Z axis [degree] (default 60): rotates the view about the Z axis.
• Zoom (default 0): increases/decreases magnification (see mapping below).
• Shift in X Direction (default 50): pans left/right.
• Shift in Y Direction (default 50): pans up/down.
• Reset View: restores the default rotation/zoom/shift.

Chart settings

• Show Points on XY Plane: draws projected “shadow” points on the XY plane.
• Show Points on YZ Plane: draws projected “shadow” points on the YZ plane.
• Show Points on XZ Plane: draws projected “shadow” points on the XZ plane.
• Point Size (for each plane): marker size for projected plane points.
• Gridlines: toggles gridlines on the three coordinate planes.
• Marker Symbol: marker shape for the main data points. Supported symbols are: Circle, Square, Diamond, Triangle, X, Plus, Star
• Marker Size: marker size for the main data points.
• Z-drop Lines: draws dashed drop lines from each data point to the XY plane.
• Show Data Point Labels: toggles display of labels from Datapoint labels.
• Label font size: font size for data labels.
• Data Point Label Position: Right / Left / Above / Below.

Note
The Marker Symbol setting applies to the main XYZ data series. Plane-projection points are rendered as circular markers.

3D Objects tab

The 3D Objects tab allows you to overlay wireframe objects on the XYZ scatter plot. The current implementation supports:

• Sphere
• Ellipsoid

Additional controls on this tab apply to all objects created from the current table:

• Number of Latitude Rings
• Number of Longitude Rings
• Points per Ring
• Color RGB

Axis Options tab

The Axis Options tab controls axis direction and axis scaling.

Axis direction

For each axis (X, Y, Z), the user can enable:

• Reverse X Axis
• Reverse Y Axis
• Reverse Z Axis

This reverses the direction in which values increase along the selected axis.

Manual axis limits

For each axis (X, Y, Z), optional manual bounds can be provided:

• Minimum
• Maximum

Rules:

• both Minimum and Maximum must be provided together;
• values must be numeric and finite;
• Minimum < Maximum.

If left blank, the axis uses automatically computed bounds.

Scale Axes

• Scale Axes rescales X/Y/Z visually so that relative raw-axis ranges are preserved in the projection.
• The scaling uses the final axis ranges after automatic bounds expansion and any manual minimum/maximum overrides.

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What the tool does

1) Data preparation and axis bounds

Let \(x_i, y_i, z_i\) be the selected values for row \(i=1,\dots,n\).

Rows with any missing value are dropped listwise.

The tool then computes axis bounds in three stages:

Stage A: automatic bounds from data

For each axis \(k\in\{x,y,z\}\), BESH Stat NG first computes:

• minimum \(m_k = \min_i k_i\)
• range \(r_k = \max_i k_i - \min_i k_i\)

using the selected XYZ data.

Stage B: expand bounds to include 3D objects

If wireframe 3D objects are attached, the automatic bounds are expanded so the full object geometry is included in the plot.

For example:

For a sphere with center \((c_x,c_y,c_z)\) and radius \(r\), the contributed bounds are:

\[ [c_x-r,\; c_x+r],\quad [c_y-r,\; c_y+r],\quad [c_z-r,\; c_z+r] \]

For an ellipsoid with semi-axes \((a,b,c)\), the contributed bounds are:

\[ [c_x-a,\; c_x+a],\quad [c_y-b,\; c_y+b],\quad [c_z-c,\; c_z+c] \]

The final automatic bounds are the union of the data bounds and all attached object bounds.

Stage C: optional manual axis limits

If the user provides manual Minimum and Maximum values on the Axis Options tab, those values override the automatically computed bounds for the selected axis.

This means the final bounds used by the renderer are:

• automatic data bounds,
• expanded to include 3D objects,
• then overridden by any manual min/max supplied by the user.

2) Normalization to a unit cube

Using the final bounds, each axis is normalized to \([-0.5,\,0.5]\):

\[ \tilde x_i = \frac{x_i - m_x}{r_x} - 0.5,\qquad \tilde y_i = \frac{y_i - m_y}{r_y} - 0.5,\qquad \tilde z_i = \frac{z_i - m_z}{r_z} - 0.5. \]

If an axis is reversed, the normalized coordinate is multiplied by \(-1\):

\[ \tilde x_i \leftarrow d_x \tilde x_i,\qquad \tilde y_i \leftarrow d_y \tilde y_i,\qquad \tilde z_i \leftarrow d_z \tilde z_i, \]

where each direction coefficient \(d_k \in \{+1,-1\}\).

This normalization makes the plot scale-invariant, while still allowing manual axis limits and axis reversal.

3) Optional axis scaling (Scale Axes)

If Scale Axes is enabled, the tool reduces visual distortion when the final raw axis ranges differ substantially.

Let \(r_{\max} = \max(r_x, r_y, r_z)\) and define scale ratios

\[ s_x = \frac{r_x}{r_{\max}},\qquad s_y = \frac{r_y}{r_{\max}},\qquad s_z = \frac{r_z}{r_{\max}}. \]

These scale ratios are computed from the final bounds actually used by the plot:

• after including 3D object bounds;
• after applying any manual axis min/max overrides.

The scaling is then applied inside the projection step (equivalent to scaling \((\tilde x_i,\tilde y_i,\tilde z_i)\) by \((s_x,s_y,s_z)\)).

If Scale Axes is disabled, each axis is normalized independently into \([-0.5,0.5]\), which can visually exaggerate variation on short-range axes.

4) 3D rotation + 2D projection

The tool uses two user-controlled rotation angles:

• \(\alpha\): X-axis rotation (Rotation X)
• \(\gamma\): Z-axis rotation (Rotation Z)

Internally, the Y-axis rotation is fixed at \(180^\circ\).

Rather than forming a full 3×3 rotation matrix and then selecting two coordinates, BESH Stat NG precomputes two projection vectors \(\mathbf{r}_1,\mathbf{r}_2\in\mathbb{R}^3\). For each point, the 2D coordinates are:

\[ u_i = \mathbf{r}_1^\top \begin{pmatrix}\tilde x_i\\\tilde y_i\\\tilde z_i\end{pmatrix}, \qquad v_i = \mathbf{r}_2^\top \begin{pmatrix}\tilde x_i\\\tilde y_i\\\tilde z_i\end{pmatrix}. \]

With \(\sin\) and \(\cos\) computed at the chosen angles (in radians), the implemented projection vectors are:

\[ \mathbf{r}_1= \begin{pmatrix} s_x\,\cos(\beta)\cos(\gamma)\\ -s_y\,\cos(\beta)\sin(\gamma)\\ s_z\,\sin(\beta) \end{pmatrix},\qquad \mathbf{r}_2= \begin{pmatrix} s_x\,[\cos(\gamma)\sin(\beta)\sin(\alpha)+\sin(\gamma)\cos(\alpha)]\\ s_y\,[-\sin(\gamma)\sin(\beta)\sin(\alpha)+\cos(\gamma)\cos(\alpha)]\\ -s_z\,\sin(\alpha)\cos(\beta) \end{pmatrix}, \]

where \(\alpha\) is Rotation X, \(\gamma\) is Rotation Z, and \(\beta\) is the fixed Y rotation (\(\beta=180^\circ\)).

Because \(\beta=180^\circ\) is fixed, the above simplifies to:

\[ \mathbf{r}_1= \begin{pmatrix} -s_x\,\cos(\gamma)\\ s_y\,\sin(\gamma)\\ 0 \end{pmatrix},\qquad \mathbf{r}_2= \begin{pmatrix} s_x\,\sin(\gamma)\cos(\alpha)\\ s_y\,\cos(\gamma)\cos(\alpha)\\ s_z\,\sin(\alpha) \end{pmatrix}. \]

5) Zoom and pan (shift)

The final chart coordinates apply a zoom multiplier and constant shifts:

\[ u_i^{\ast} = (u_i + \delta_x)\,z_{\text{int}},\qquad v_i^{\ast} = (v_i + \delta_y)\,z_{\text{int}}. \]

The internal mapping from the UI controls is:

• Zoom \(z\) → \(z_{\text{int}} = \bigl(1 + z/50\bigr)^2\)
• Shift X \(s\) → \(\delta_x = s/100 - 0.5\)
• Shift Y \(t\) → \(\delta_y = t/100 + 0.5\)

These transforms are also applied to the cage, tick marks, gridlines, and plane projections so the entire scene moves consistently.

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Plane projections (shadow points)

When enabled, “shadow points” are created by replacing one coordinate with the minimum face of the cube (normalized value \(-0.5\)) and projecting the result.

For example, XY-plane projections use \((\tilde x_i,\tilde y_i, -0.5)\). Likewise:

• XZ plane: \((\tilde x_i, -0.5, \tilde z_i)\)
• YZ plane: \((-0.5, \tilde y_i, \tilde z_i)\)

Each plane’s points are drawn as a separate Excel series with the chosen Point Size.

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Z-drop lines

When Z-drop Lines is enabled, the tool draws a dashed line from each data point down to its XY-plane projection.

Implementation detail: Excel custom minus error bars are used to create these lines.

Let \(v_i^{*}\) be the projected Y coordinate of the full 3D point, and \(v_{i,\text{XY}}^{*}\) the projected Y coordinate of its XY-plane shadow point. The error-bar magnitude is:

\[ e_i = v_i^{*} - v_{i,\text{XY}}^{*}. \]

Excel draws a vertical (screen-space) dashed segment of length \(e_i\) from each marker, producing a consistent “drop” effect.

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Tick marks and gridlines

Tick values

For each axis, the tool chooses a “nice” tick step based on one-fifth of the raw range.

Let

\[ f_k=\frac{r_k}{5}. \]

It then sets the number of decimal digits to

\[ n_{\text{digits}} = 1 - \left\lfloor \log_{10}(|f_k|) + 1 \right\rfloor \]

and computes

\[ \text{step}_k = \operatorname{round}(f_k,\; n_{\text{digits}}). \]

The first tick is the first multiple of \(\text{step}_k\) above the minimum:

\[ \text{first}_k = \text{step}_k\,\Bigl(\lfloor m_k/\text{step}_k\rfloor + 1\Bigr). \]

Tick labels are then placed along each axis in the drawn cage.

Gridlines

If Gridlines is enabled, the tool draws dashed gridlines on the XY, XZ, and YZ planes using the same normalized coordinate system and the current projection.

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Output

The tool creates an Excel Chart sheet named XYZ 3D (Excel may append a suffix if a sheet of that name already exists). The chart contains:

• the 3D bounding cage and axis labels;
• tick marks and numeric tick labels;
• optional plane-projected points;
• main 3D-projected data points (colored by Group ID if provided);
• optional wireframe 3D objects (sphere / ellipsoid);
• optional point labels;
• optional Z-drop lines.

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Notes and tips

• If you provide a Group ID, each group is drawn as a separate Excel series and appears in the legend as Group_<value>.
• If Show Data Point Labels is enabled but no label range is supplied, no labels are shown.
• For large \(n\), plotting may be slow because each feature (cage, ticks, gridlines, planes, groups) adds Excel series and formatting.

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Animated GIF workflow

The Animated Gif feature creates an animated 3D scatter plot by redrawing the chart from a sequence of view settings defined in an input table.

The current implementation does both steps:

1. exports individual frame GIFs;
2. combines those frames into a final animated GIF.

UI location

The workflow is controlled from:

• Animated Gif Inputs
• Animated Gif button

on the main Input tab.

Animated Gif Inputs format

The Animated Gif Inputs range must be a rectangular range with 6 numeric columns and one row per frame.

Column definitions (left → right)

1. Rotation X [degree]
   Rotation angle about the X axis.
   Valid range: \([0, 360]\)

2. Rotation Z [degree]
   Rotation angle about the Z axis.
   Valid range: \([0, 360]\)

3. Zoom
   Zoom control value.
   Must be >= 0.

4. Shift in X Direction
   Horizontal pan value.
   Valid range: \([0, 100]\)

5. Shift in Y Direction
   Vertical pan value.
   Valid range: \([0, 100]\)

6. Delay [ms]
   Per-frame delay in milliseconds.
   Must be >= 0.

Notes

• One row = one frame.
• Frame indices start at 0.
• If you want a constant view, repeat the same row for multiple frames.
• Smooth animations are typically created by changing one parameter gradually across rows.

Output path and frame folder

When you click Animated Gif, the tool prompts you for the final output .gif file path.

If you choose:

• C:\Plots\xyz_spin.gif

the tool creates a working folder:

• C:\Plots\xyz_spin_frames\

and stores intermediate frames there as:

• 0.gif, 1.gif, 2.gif, …, n-1.gif

where n is the number of rows in Animated Gif Inputs.

After all frames are exported, the tool combines them into the final animated GIF at the selected output path.

Example

A simple rotation-only animation can be created by keeping Zoom and Shift fixed while changing Rotation X or Rotation Z over rows.

For example:

• Rotation X: 110, 120, 130, …
• Rotation Z: 60 (constant)
• Zoom: 0 (constant)
• Shift in X Direction: 50 (constant)
• Shift in Y Direction: 50 (constant)
• Delay: 110 ms (constant)

Example output

Tips for smooth animations

• Prefer smaller rotation steps (e.g. 2–5 degrees per frame) for smoother motion.
• Keep Zoom and Shift fixed while you debug rotation.
• If you see clipping or a jump in the view, reduce the step size or increase the number of frames.
• Use a constant Delay [ms] (e.g. 80–150 ms) during testing and adjust later for the desired playback speed.

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3D Objects (wireframe shapes)

The 3D Objects tab allows you to overlay one or more wireframe shapes (e.g., spheres and ellipsoids) on top of the XYZ 3D scatter plot.

3D Objects Specification (range)

3D Objects Specification must be a rectangular table (Excel range) where each row defines one object.

• Column 1 (Type): object type (text)
• Column 2–4: center coordinates X, Y, Z (numeric; same units as your plotted data)
• Remaining columns: parameters depend on the object type

Supported types (case-insensitive), including convenience aliases:

• Sphere: sphere, s, wire_sphere, wiresphere
• Ellipsoid: ellipsoid, e, wire_ellipsoid, wireellipsoid

Mixed table format (recommended)

Because one Excel range must have a single fixed width, the recommended mixed-format table uses 7 columns so it can contain both spheres and ellipsoids:

Col	Name	Used by
1	type	sphere / ellipsoid
2	X	sphere / ellipsoid
3	Y	sphere / ellipsoid
4	Z	sphere / ellipsoid
5	diameter or diameterX	sphere / ellipsoid
6	diameterY	ellipsoid (ignored for sphere rows)
7	diameterZ	ellipsoid (ignored for sphere rows)

Sphere rows use column 5 as diameter and may leave columns 6–7 blank.
Ellipsoid rows use columns 5–7 as diameterX, diameterY, diameterZ.

Example (mixed objects in one range):

type	X	Y	Z	d / dx	dy	dz
sphere	10	5	3	8
ellipsoid	0	0	0	10	6	4

Validation rules

The tool validates the specification range before creating objects:

• Type must be one of the supported values (see above).
• X, Y, Z must be numeric and finite.
• Diameters must be numeric, finite, and >= 0.
• Objects with diameter = 0 (or ellipsoid with any diameterX/Y/Z = 0) are treated as degenerate and are skipped.

Interaction with axis bounds

By default, the plot automatically expands X/Y/Z bounds to include the full extent of attached 3D objects, so wire spheres and ellipsoids are fully visible even when they extend beyond the raw data cloud.

If manual axis Minimum/Maximum values are supplied on the Axis Options tab, those manual bounds override the automatic object-expanded bounds for the affected axis.

Wireframe density and appearance

The following settings apply to the wireframe rendering:

• Number of Latitude Rings: number of “horizontal” rings (more = denser wireframe).
• Number of Longitude Rings: number of “vertical” meridians (more = denser wireframe).
• Points per Ring: number of points used to approximate each ring (more = smoother curves).
• Color RGB: wireframe line color (applies to objects drawn with the current settings).

Tip
Higher ring counts and points-per-ring can look better but may slow down rendering because each object is drawn as line series data in Excel.

How the inputs are used (math)

All 3D objects are defined in the same coordinate system as your plotted data.
Let the object center be:

\[ \mathbf{c} = (c_x, c_y, c_z) \]

Sphere

For a sphere, the input diameter \(d\) is converted to the radius:

\[ r=\frac{d}{2} \]

A sphere is the set of points \((x,y,z)\) satisfying:

\[ (x-c_x)^2 + (y-c_y)^2 + (z-c_z)^2 = r^2 \]

For rendering a wireframe, the sphere is parameterized using latitude/longitude angles:

• latitude \(\varphi \in \left[-\frac{\pi}{2}, +\frac{\pi}{2}\right]\)
• longitude \(\theta \in [0, 2\pi]\)

\[ \begin{aligned} x(\theta,\varphi) &= c_x + r\cos\theta\cos\varphi \\ y(\theta,\varphi) &= c_y + r\sin\theta\cos\varphi \\ z(\theta,\varphi) &= c_z + r\sin\varphi \end{aligned} \]

Wireframe rings are drawn as:

• Latitude rings: fix \(\varphi\) and sweep \(\theta\) from \(0\) to \(2\pi\)
• Longitude rings: fix \(\theta\) and sweep \(\varphi\) from \(-\frac{\pi}{2}\) to \(+\frac{\pi}{2}\)

Ellipsoid

For an ellipsoid, the inputs diameterX, diameterY, diameterZ are converted to semi-axes:

\[ a=\frac{d_x}{2}, \quad b=\frac{d_y}{2}, \quad c=\frac{d_z}{2} \]

An axis-aligned ellipsoid is the set of points satisfying:

\[ \frac{(x-c_x)^2}{a^2} + \frac{(y-c_y)^2}{b^2} + \frac{(z-c_z)^2}{c^2} = 1 \]

The wireframe is parameterized similarly:

\[ \begin{aligned} x(\theta,\varphi) &= c_x + a\cos\theta\cos\varphi \\ y(\theta,\varphi) &= c_y + b\sin\theta\cos\varphi \\ z(\theta,\varphi) &= c_z + c\sin\varphi \end{aligned} \]

As with the sphere:

• Latitude rings fix \(\varphi\) and sweep \(\theta\)
• Longitude rings fix \(\theta\) and sweep \(\varphi\)

Ring counts and smoothness

• Number of Latitude Rings controls how many distinct latitude values \(\varphi\) are used (excluding the poles).
• Number of Longitude Rings controls how many distinct longitude values \(\theta\) are used.
• Points per Ring controls how finely each ring is sampled (how many \(\theta\) or \(\varphi\) steps per ring), affecting curve smoothness.

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See also

• Scatter Plot Matrix
• Home